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OF  THI-: 

UNIVERSITY  OF  CALIFORNIA. 

\     zAccessioiis  No.  (:;i^^O  f.     CLns  No. 

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Number  and  its  Algebra 


SYLLABUS  OF  LECTURES  ON   THE  THEORY 
OF  NUMBER  AND  ITS  ALGEBRA 


INTRODUCTORY  TO   A 


COLLEGIATE  COURSE  IN  ALGEBRA 


BY 

ARTHUR  LEFEVRE,  C.E. 

Instructor  in  Pure  Mathematics,  University  of  Texas 


university) 


BOSTON,  U.S.A. 

D.  C.  HEATH  &  CO.,  PUBLISHERS 

1896 


Copyright,  1896, 
By  Akthue  Lefevre. 


C.  J.  Peteks  &  SON,  Typoqbaphees,  Boston. 


S.  J.   PaEKHTI.I,   &  Co.,    PBINTEB8. 


DEDICATED 
THE  TEACHERS  OF  MATHEMATICS 

IX   THE 

C0M:\I0X  SCHOOLS. 


(trNIVERSITT^ 


(.svs^f 


•0NI 


G^UFORN. 


CONTENTS. 


CHAPTER  PAGE 

Introduction ' '.     .    .  5 

I.    Pbimaky  Number 19 

II.     Counting , 21 

III.  Some  Fundamental  Theory 24 

IV.  Notation 25 

Y.    Algebra '. 31 

VI.     Calculation 37 

VII.     Primary  Number.  —  Numerical  Operations    .     .  42 

YIII.     Devices  of  Computation 58 

IX.     First  Extension  of  the  Number-Concept.     .     .  61 
Eatio.  —  Fractions.  — Surds. 

X.     Significance  of  Operations,  and  Special  Opera- 
tional   Devices    Appropriate    to    the    First 

Extension  of  the  Number-Concept     ....  65 

XI.     Final    Extension    of    the    Number-Concept.  — 

Principle  of  Continuity 73 

xii.  signlificance  and  efficacy  of  numerical  op- 
erations under  the  ultimate  concept.  — 
Zero,  Infinity.  —  Negative,  Neomonic,  Com- 
plex Number 85 

XIII.  Measurement 125 

XIV.  Mathematics 134 

XV.     Some  Theorems  and  Problems 146 


J 


INTRODUCTION. 


"  The  scientific  part  of  Arithmetic  and  Geometry  would  be  of  more 
use  for  regulating  the  thoughts  and  opinions  of  men  than  all  the  great 
advantage  which  Society  receives  from  the  practical  application  of 
them :  and  this  use  cannot  be  spread  through  the  Society  by  the  prac- 
tice ;  for  the  Practitioners,  however  dextrous,  have  no  more  knowledge 
of  the  Science  than  the  very  instruments  with  which  they  work.  They 
have  taken  up  the  Rules  as  they  found  them  delivered  down  to  them  by 
scientific  men,  without  the  least  inquiry  after  the  Principles  from  which 
they  are  derived :  and  the  more  accurate  the  Rules,  the  less  occasion 
there  is  for  inquiring  after  the  Principles,  and  consequently,  the  more 
difficult  it  is  to  make  them  turn  their  attention  to  the  First  Principles ; 
and,  therefore,  a  Nation  ought  to  have  both  Scientific  and  Practical 
Mathematicians."  — James  Williamson,  Elements  of  Euclid  with  Dis- 
sertatio7is,  Oxford,  1781. 

The  preceding  arraignment  is  nearly  as  pertinent  to-day 
in  this  country  as  it  was  in  England  more  than  a  centui-y 
ago.  But  so  far  as  Geometry  is  concerned  blame  no  longer 
rests  with  the  scientific  mathematicians.  Their  investiga- 
tions of  First  Principles  have  not  only  furnished  us  with 
Euclid  in  his  purity,  but  have  developed  entirely  new 
and  equally  consistent  geometries,  under  postulates  alter- 
nate to  Euclid's  petition  of  the  angle-sum  of  a  rectilineal 
triangle.  Thus  has  been  fulfilled  what  must  at  least  have 
opened  up  as  dim  vistas  to  Euclid's  mind  when  he  dis- 
cerned the  necessity  for  assuming,  or  petitioning  as  the 
old  geometers  called  it,  his  indemonstrable  postulate.* 

*  Called  variously  the  5th  postulate,  or  the  11th  or  12th  axiom. 


b  INTEODUCTION". 

Still  further,  scientific  mathematicians,  besides  offering 
the  true  Euclid  in  available  text-books  with  desirable  ad- 
ditions and  extensions,  have  corrected  several  errors  in 
definitions  and  demonstrations  which  constituted  the  sole 
blemishes  in  the  most  perfect  work  ever  performed  by  a 
single  man.  There  is  no  longer  good  excuse  for  teachers 
choosing  texts  which  present  the  postulate  as  a  common 
notion  or  axiom;  to  say  nothing  of  such  as  baldly  omit 
the  whole  doctrine  of  ratios  and  proportionality.  There  is 
a  momentous  difference  between  ratios  and  fractions,  and 
text-books  which  present  a  proportion  simply  as  an  equality 
of  fractions  have  set  up  a  miserable  cause  of  stumbling. 
They  consider  "merely  a  special  case  of  no  importance, 
whose  only  excuse  for  existence  lies  in  the  general  case 
omitted."  *  Incommensurability  is  the  rule,  commensura- 
bility  the  exception. 

On  the  other  hand,  when  we  consider  Arithmetic  and 
Algebra  the  cap  of  censure  fits  the  other  head.  If  our 
scientific  mathematicians  have  furnished  satisfactory  text- 
books in  these  subjects,  I  am  not  acquainted  with  them. 
All  of  us  who  are  teaching  mathematics  must  agree  with 
good  old  Williamson  when  he  complains,  in  the'dissertation 
already  quoted,  that  he  found  it  more  difficult  "  to  make 
a  rational  arithmetician  than  an  enlightened  geometer." 

Let  me  hasten  to  say  that  the  apparently  controversial 
tone  of  this  preface  springs  from  no  polemical  spirit.  I 
approach  the  task  I  have  set  myself  with  utmost  modesty; 
nay,  oppressed  by  a  sense  almost  of  presumption  in  at- 
tempting to  clarify  what  so  many  have  left  confused.  But 
so  sorely  needed  is  a  successful  accomplishment  of  what  I 


*  Catalogue  Univ.  of  Texas,  1891-1892. 


INTRODUCTION.  7 

attempt,  that  an  honest  effort  needs  no  apology.  I  wish 
also  to  explain  that  the  present  treatment  takes  its  form 
from  the  immediate  practical  aim  in  view ;  viz.,  that  of  a 
syllabus  for  a  rapid  review  of  such  ground  of  arithmetic 
and  algebra  as  will  best  prepare  for  the  study  of  what 
goes  in  our  curricula  by  the  name  of  "  higher  algebra," 
with  special  adaptation  to  the  needs  of  that  large  portion 
of  my  classes  who  are  taking  the  course  in  order  to  qualify 
as  teachers  in  the  public  schools. 

I  write  this  Introduction,  and  dedicate  the  little  work  to 
the  teachers  in  the  common  schools,  however,  in  the  hope 
of  attaining  a  wider  usefulness,  in  the  way  of  awakening 
in  some  Practical  Mathematician  a  desire  ""to  make  ra- 
tional arithmeticians  "  of  the  youths  whose  studies  he  is 
directing.  It  is  proper  to  explain  still  further  that,  work- 
ing away  from  any  great  library,  I  have  been  compelled 
to  prepare  this  matter  for  printing  without  having  time  to 
procure  a  few  published  works  which  I  would  like  to  see 
before  committing  myself  to  publication. 

I  must  not  be  understood  as  advancing  anything  new  to 
mathematicians,  though  I  know  of  no  English  text-book 
which  consistently  expounds  and  maintains  the  theories 
of  number  and  algebra  here  presented.  The  work  is  ad- 
dressed, not  to  mathematicians,  but  to  inquiring  students 
and  teachers.  A  sound  doctrine  of  number  and  its  algebra 
seems  to  be  left  by  our  text-books  to  chance  inference,  or 
deferred  to  stages  seldom  reached  in  undergraduate  courses 
of  study.  A  straightforward  development,  comprehensible 
by  beginners,  of  the  number  concept  would  be  of  immense 
service  in  mathematical  instruction. 

For  six  years  I  have  given  my  classes  the  substance  of 
this  syllabus  as  the  best  explanation  I  could  offer  of  dififi- 


8  INTRODUCTION. 

culties  which  could  not  honestly  be  avoided.  In  July, 
1894,  I  read  in  the  current  issue  of  the  Monist  an  article 
by  Hermann  Schubert,  writing  in  Hamburg,  on  Monism  in 
Arithmetic,  enunciating  a  unifying  principle  which  he  called 
the  Principle  of  No  Exception,  referring  it  originally  to 
Hankel.  Of  course  some  such  principle  is  more  or  less 
clearly  in  the  mind  of  every  student  of  mathematics,  but 
having  never  read  Hankel's  own  statement,  I  cannot  say 
whether  his  Prlticlple  of  Permanence  is  substantially  iden- 
tical with  the  developing  principle  I  set  forth,  or  rather 
in  line  with  the  notion  of  algebra  as  "the  science  which 
treats  of  the  combinations  of  arbitrary  signs  and  symbols, 
by  means  of  defined,  though  arbitrary,  laws,"*  —  the  view 
of  the  famous  Dean  of  Ely,  and  the  long  line  of  algebraists 
of  whom  he  is  the  prototype.  The  bare  statements  of 
such  a  principle  from  radically  different  standpoints  might 
be  confusingly  similar  to  one  not  fully  alive  to  the  fun- 
damental variance.  For  example,  in  Schubert's  statement 
of  his  Principle  of  No  Exception,  I  recognized  what  I 
conceive  to  be  a  somewhat  inadequate  expression  of  the 
postulate  I  had  called  the  Principle  of  Continuity  (I  still 
prefer  this  name  as  pointing  with  direct  emphasis  to  its 
cardinal  outgrowth  —  the  conception  of  number  as  continu- 
ous), whereas  in  the  next  preceding  issue  of  the  same 
journal  he  is  at  utter  variance  with  me  in  declaring  that, 
"all  numbers,  excepting  the  results  of  counting,  are  and 
remain  mere  symbols,  nothing  but  artificial  inventions  of 
mathematicians." 


*  Peacock's  Report  on  the  Recent  Progress  and  Present  State  of  cer- 
tain branches  of  Analysis,  in  the  British  Association  Report  for  1833, 
p.  195.  Cf.  also  Peacock's  Treatise  on  Algebra,  1830,  republished  and 
enlarged  in  1842. 


INTRODUCTION.  9 

In  the  article  above  referred  to,  Schubert  claims  that  in 
his  System  of  Arithmetic,  Potsdam,  1885,  he  ''  was  the  lirst 
to  work  out  the  idea  referred  to  fully  and  logically,  and  in  a 
form  comprehensible  for  beginners  ;  "  although  it  had  been 
previously  expressed  by  Grassman,  Hankel,  E.  Schroeder, 
and  Kronecker.  Such  is  the  bibliography  of  this  special 
presentation  of  the  subject,  so  far  as  I  am  aware,  not  to 
mention  Dr.  Halsted's  Number,  Discrete  and  Continuous,* 
whose  title  promises  a  treatment  of  this  subject,  but  which 
remains  a  fragment,  dealing  only  with  discrete  number  — ■ 
what  I  have  called  Primary  Number.  Should  I  be  able  to 
spur  Dr.  Halsted  to  a  completion  of  this  work  I  shall  not 
have  written  in  vain. 

Of  course,  Hankel's  principle  must  be  expounded  in  his 
Theorie  der  complexen  Zahlansysteme,  Leipzig,  1867 ;  but 
I  am  yet  ignorant  of  the  specific  publications  of  the  other 
authors  named,  except  that  in  Zeller's  jubilee  work  the 
matter  is  referred  to  in  an  essay  by  Kronecker. 

On  the  other  hand,  the  theory  here  advocated  must  not 
be  deemed  retrogressive,  and  referred  to  such  writers  as 
Frend,t  who,  though  he  very  philosophically  maintains 
that,  since  algebra  has  its  origin  and  termination  in  arith- 
metic, it  cannot  be  considered  independent,  and  fairly 
enough  regards  algebra  as  "  the  science  which  teaches  the 
'  general  properties  and  relations  of  numbers,"  yet  ends  by 
practically  throwing  the  greater  part  of  the  science  of 
number  overboard,  in  rejecting  all  algebraic  forms  which 
do  not  agree  with  his  undeveloped  concept  of  number. 

My  theme  may  be  regarded  as  the  underlying  harmony 

*  Preface  and  four  chapters  (22  pp.)  in  Scientiae  Baccalaureus,  June, 
1891. 

t  Algebra,  1796. 


10  INTRODUCTION. 

of  the  great  makers  of  analytical  mathematics,  —  and  my 
purpose,  as  an  attempt  to  present  to  beginners  fundamental 
theory  commonly  left  for  the  speculations  of  the  most 
advanced. 

Number  is  such  a  perfect  and  typical  abstraction  that 
it  is  difiScult  to  see  how  a  man  who  has,  to  use  Newton's 
phrase,  "  in  philosophical  matters  a  competent  faculty  of 
thinking,"  could  ever  associate  the  terms  concrete  and  num- 
ber ;  nevertheless  this  confusion  muddles  many  popular 
text-books.  The  question  hardly  requires  or  admits  of 
argument.  Since  it  is  a  vicious  habit  rather  than  an  illogi- 
cal deduction  which  is  to  be  combated,  good-tempered  ridi- 
cule is  perhaps  the  only  fit  rejoinder.  In  this  spirit  may 
I  be  permitted  to  relate  an  anecdote  ?  Some  years  ago  at 
the  University  of  Virginia  the  Professor  of  Mathematics 
assigned  several  problems  to  be  worked  upon  the  black- 
boards by  members  of  the  Junior  Class.  To  one  he  gave  a 
problem  concerning  the  number  of  oranges  in  a  pyramidal 
pile  of  stated  proportions.  After  expounding  the  error  or 
propriety  of  the  solutions  of  some  of  the  other  problems, 
the  turn  of  the  orange  problem  came.  The  student  stood 
proudly  beside  his  mechanically  correct  solution.  "  Well, 
Mr.  Blank,"  exclaimed  the  Professor,  "  how  many  apples 
did  you  find  ?  "  A  look  of  consternation  overspread  the 
youth's  countenance.  With  a  gesture  of  impatient  annoy- 
ance he  swept  the  erasing  brush  over  the  figures  his  chalk 
pencil  had  traced  :  ''  Oh,"  said  he,  "  I  thought  you  said 
oranges  !  "  In  all  seriousness,  the  text-books  we  have  all 
been  abused  by,  expounding  "  concrete  numbers,"  solemnly 
cautioning  against  confusion  of  multiplicand  and  multi- 
plier, divisor  and  quotient,  and  unallowable  combinations  of 
the  terms  of  a  numerical  proportion,  are  quite  as  ridiculous 


INTRODUCTION.  11 

as  our  hero  of  the  oranges.  He  displayed  at  least  one 
virtue,  —  consistency.  Such  questions,  however,  though  of 
great  practical  importance  to  the  efficiency  of  our  elemen- 
tary schools,  present  no  real  difficulties.  A  little  knowledge 
of  psychology  and  mathematics  will,  if  attention  he  called 
to  the  question,  correct  mistaken,  and  develop  inchoate 
concepts  of  Primary  I^umber.  A  far  more  difficult  matter 
remains  —  to  attain  for  ourselves,  and  to  lead  our  pupils  to 
attain,  a  rational  concept  of  number  as  continuous,  a  con- 
cept absolutely  essential  to  modern  mathematics,  and  now 
universally  assumed  as  a  fact,  —  implicitly  so  assumed,  even 
when  explicitly  denied.  It  is  also  necessary  to  pass  beyond 
the  great  step  already  made  by  Newton,  who  discerns  the 
continuity  of  number,  but  leaves  it  only  ''  triplex  "  :  "  Est- 
que  (Humerus)  triplex  ;  integer,  f  ractus,  et  surdus  :  Integer 
quem  unitas  metitur,  Fractus  quern  unitatis  pars  submulti- 
plex  metitur,  et  Surdus  cui  unitas  est  incommensurabi- 
lis  "  *,  with  the  implied  limits  zero  and  infinity.  Xewton 
also  recognized  qualitative  distinctions,  positive  and  nega- 
tive, but  the  consequent  neomonic  (so-called  "  imaginary  ") 
and  complex  numbers  remain  to  be  assimilated.  I  must 
return,  however,  to  notice  a  uniquely  erroneous  view  of 
primary  number  presented  in  the  last  issue  of  the  Inter- 
national Education  Seriks,  The  T'sycliology  of  Number, 
by  James  A.  McLellan,  A.M.,  LL.D.,  Principal  of  the  On- 
tario School  of  Pedagogy,  Toronto,  and  John  Dewey,  Ph.D., 
Head  Professor  of  Philosophy  in  the  University  of  Chi- 
cago, edited  like  all  of  the  series  by  W.  T.  Harris,  U.  S. 
Commissioner  of  Education. 

The  astounding  thesis  is  maintained  that  number  is  not  a 

*  Arithmetica  universalis:  quoted  fi-oiu  Halsted's  Number,  Discrete 
and  Continuous. 


12  INTRODUCTION. 

magnitude,  does  not  possess  quantity  at  all,  and  that  "  no 
number  can  be  multiplied  or  divided  into  parts."  *  The 
authors  vehemently  assert  that  we  might  as  well  talk  of 
any  absurdity  "  as  to  talk  of  multiplying  a  number."  f 
It  is  much  to  be  regretted  that  a  Avork  of  such  prestige 
should  merely  shift  the  misconception  of  concreteness  from 
numbers  to  the  subjects  of  calculation,  which  we  are  told 
to  believe  are  never  numbers  at  all.  Number  is  most  em- 
phatically shown  to  be  "  purely  abstract,"  |  yet  multipli- 
cation is  claimed  to  be  only  of  concretes.  It  is  nonsense, 
we  are  told,  to  think  of  multiplying  six  by  four ;  you  can 
only  multiply  six  inches,  six  oranges,  by  four.  Of  course, 
that  numbers  are  multiplied  is  a  fact,  a  fact  that  psy- 
chology may  explain,  but  can  in  no  wise  question.  After 
repeatedly  insisting  upon  "  the  absurdity  of  multiplying 
pure  number  or  dividing  it  into  parts,"  §  the  authors  admit 
without  comment,  and  in  seeming  hesitation,  '•'  of  course, 
in  all  mathematical  calculations  we  ultimately  operate  with 
pure  symbols."  ||  What  are  these  "  pure  symbols  "  ?  What 
can  they  be  in  arithmetic  but  the  pure  numbers  them- 
selves ?  It  woiild  be  an  error,  shared  by  many  algebraists, 
to  conceive  algebra  as  lacking  specific  content  —  as  operat- 
ing with  '''  pure  symbols,"  whatever  that  may  mean.  The 
chapter  on  the  Psychical  Nature  of  Nurnber  is  admirable, 
and  I  gratefully  invoke  its  corroboration  of  what  will  be 
found  in  my  syllabus  on  the  subject;  but  that  upon  the 
Origin  of  Number,  though  very  acute  in  tracing  the  de- 
pendence of  measurement  upon  "  adjustment  of  activity," 
seems  to  me  mistaken  in  finding  the  origin  of  number  in 

*  Psychology  of  Number,  p.  70.        §  Ih.,  p.  71,  foot-uote. 
t  lb.,  p.  70.  II  lb.,  p.  71. 

X  lb.,  p.  69. 


INTRODUCTION.  13 

measurement.  Measurement  is  not  the  source  of  the  con- 
cept of  number,  but  a  stimulation  to  clarify  and  develop 
the  concept;  and  this  is  what  the  facts  cited  really  show. 
The  primary  concept  of  number,  as  so  correctly  defined 
in  the  preceding  chapter,  is  prerequisite  to  any  attempt  at 
measurement.  The  savage  referred  to  needs  the  concept 
that  the  length  of  his  arrow  is  some  number  of  hand- 
breadths  before  he  can  attempt  to  discover  how  many. 
And  long  before  this  he  has  learned  to  recognize  a  small 
group  of  objects  as  a  "vague  whole,"  and  to  "discriminate 
the  distinct  individuals,"  i.e.,  in  the  very  terms  employed 
to  define  number,  the  concept  originated  before  any  meas- 
urement became  possible.  Nor,  in  truth,  does  number  ori- 
ginate in  counting,  as  so  commonly  asserted;  and  for  a 
like  reason,  viz.,  the  concept  of  some  number  must  pre- 
cede any  device  for  naming  or  anywise  specializing  it. 
The  position  that  number  has  its  origin  in  measurement 
cannot  seek  strength  from  the  procession  toward  the  ab- 
solute of  Hegel's  ascending  categories,  quality,  quantity 
(including  number,  as  "  quantum  in  its  complete  spe- 
cialization "),  measure  {das  Maass),  essence ;  *  for  Hegel's 
Maass,  i.e.,  "  qualitative  quantity  or  measure,"  f  is  a  very 
different  matter  from  Dr.  Dewey's  measurement,  in  fact, 
it  seems  very  nearly  the  same  as  number  according  to  the 
growing  insight  of  modern  mathematics. 

Having  mentioned  Hegel,  it  is  proper  to  remark  that  we 
are  just  now  being  reminded  on  every  side  —  Helmholtz 
not  long  ago  admonished  us  —  that  students  of  science  are 
frequently  driven  by  the  very  logic  of  their  subjects  into 

*  The  Logic  of  Hegel,  translation,  Wallace,  p.  192  ("quantum,  i.e., 
limited  quantity,"  p.  190). 
t  lb.,  p.  200. 


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14  INTRODUCTION. 

the  regions  of  philosophy.  A  vital  service  will  be  rendered 
any  serious  student,  should  he  be  led  to  consider,  either  at 
first  or  at  second  hand,  the  aperqus  of  Hegel  and  the  power 
of  his  method.  Without  doubt,  Hegel  has  pointed  out  the 
true  way  of  logic,  if  he  did  not  always  follow  that  path. 
His  system  is  far  from  fully  elaborated ;  much  is  tentative, 
doubtless  much  mistaken ;  but  the  fundamental  business  of 
logic  (in  the  Hegelian  sense)  must  remain  as  he  appointed 
it,  a  criticism  of  the  very  terms  of  scientific  and  ordinary 
thought ;  nor  is  a  better  method  than  his  dialectic  likely 
to  be  discovered.  Dr.  Harris  (who  — inice  —  writes  better 
than  he  edits)  has  done  our  nation  substantial  service  in 
his  HegeVs  Logic,  by  condensing,  elucidating,  and  compara- 
tively popularizing  a  work  of  prime  importance  in  the  prog- 
ress of  human  thought.  If  any  reader  has  perused  even 
the  preface  of  this  book,  I  beg  him  to  recall  the  suggestive 
and  tonic  way  in  which  Dr.  Harris  recounts  his  gradual 
and  successive  attainment  of  various  ''insights"  in  mat- 
ters philosophical,  and  to  find  encouragement  therefrom 
should  he  stumble  at  the  development  of  the  concept  of 
primary  number  which  mathematics  imperatively  demands. 
D'Alembert's  advice  to  beginners  in  the  differential  calcu- 
lus was  "  allez  en  avant  et  la  foi  vous  viendra." 

It  remains  only  to  say  that  in  this  attempt  to  elucidate 
a  unifying  principle  of  number,  and  to  display  the  nature 
of  any  algebra,  I  have  kept  in  mind  the  capacity  of  ''fresh- 
man "  students,  and  have  avoided  all  reference  to  ultimate 
categories,  psychological  or  ontological. 

I  am  well  aware  that  there  are  other  avenues  of  approach 
to  the  thesis  here  maintained,  —  that  "  various  new  mathe- 
matical conceptions  have  been  employed  by  Weierstrass, 
G.  Cantor,  and  Dedekind  in  establishing  three  independent 


DSfTIlODUCTION.  15 

and  equally  cogent  theories  which,  should  prove  the  conti- 
nuity of  number  icithout  horroiving  it  from  space,'"  *  to  say 
nothing  of  such  theories  (e.g.,  Fine's  Number- System)  as 
are  '•'  content  to  get  continuity  from  the  line."  f  Something 
tangible  for  beguaners  is  a  great  desideratum.  My  aim  is 
practical,  and  it  may  be  claimed  that  even  if  difficulties 
have  not  been  surmounted,  or  obscurities  illuminated,  they 
have  at  least  been  reduced  to  one  clear-cut  postulate.  The 
student  may  take  stock  of  his  knowledge,  and  rationally 
prosecute  his  studies,  even  though  he  consider  a  gratuitous 
assumption  left  in  the  rear ;  "  la  foi  viendra." 

To  such  as  may  condemn  the  occasional  analogical  sug- 
gestions, and  references  to  general  philosophy  in  this 
treatise,  as  unbecoming  the  proprieties  of  the  severest  of 
the  sciences,  I  would  beg  to  reply  that  the  style  of  an 
attempt  to  explain  how  mathematics  came  to  be,  and  what 
it  is,  of  an  effort  to  lead  those  who  sit  in  darkness  to  form 
the  concepts  with  which  mathematics  deals,  ought  not  to 
be  judged  by  the  standards  of  the  severe  and  self-contained 
procedure  of  the  full-fledged  science.  My  subject  soon 
enters,  but  begins  outside  of  mathematics ;  nor  is  it  ped- 
antry to  be  philosophical  in  explaining  the  fundamental 
concepts  of  any  science.  It  would  be  impertinent  to  be 
anything  else, 

I  would  also  deprecate  any  charge  of  presumption  on 
account  of  several  innovations  in  terminology.  I  am  fully 
aware  that  reformation  must  come,  if  at  all,  from  powerful 
leaders ;  but  it  seemed  appropriate  in  a  work  of  pedagogi- 


*  Number,  Discrete  and  Continuous,  George  Brace  Halsted,  Preface. 
The  italics  are  mine.  So  far  as  I  know  no  one  of  these  demonstrations 
has  appeared  in  English. 

t  /i. 


16 


INTEODUCTIOlSr. 


cal  intent  to  point  out  certain  misnomers,  and  even  to 
"practice  what  I  preach."  No  confusion  can  arise  from 
using  neomonic  and  protomonic  for  '•'  imaginary  "  and  "  real  " 
etc. ;  and  those  who  deem  the  current  terms  consecrated 
by  the  usage  of  the  great  geometers  who  have  made  the 
science,  may  ignore  the  suggestions. 

My  practical  aim  must  explain  the  apparently  arbitrary 
intrusion  of  detail,  especially  in  the  final  chapter.     In  this 
final  chapter,  it  should  be  said,  I  have  drawn  freely  from 
Professor    Chrystal's    Text  Book  of  Algebra,    Adam    and 
Charles   Black,  Edinburgh,  1886.      Such  points   only  are 
touched  upon  as  have  been  shown  by  experience  to  bear 
directly  on  the  preparation  proper  to  our  freshman  course 
in   algebra.     Especially  in  freedom  of    arrangement   and 
allusion,  some  familiarity  with  the  subject-matter  is  pre- 
supposed;   but   the   knowledge   assumed  need  be  neither 
great  nor  accurate  in  order  to  comprehend  what  is  presented. 
The  vague  acquaintance  with  terms  and  processes  possessed 
by  the  ordinary  high-school  graduate  has  sometimes  war- 
ranted the  projection  of  a  particular  discussion  beyond  the 
parallel  development  of  cognate  topics  in  a  way  which  would 
not  be  admissible  in  teaching  children.     My  classes  stand 
upon  a  vantage  ground  whence  it  is  permitted  to  look  both 
forward  and  backward,  and  so  at  last  to  command  a  really 
comprehensive  view.     A  teacher  should  never  forget,  how- 
ever, that  at  every  stage  there  should  be  an  index  pointing 
upward.     Any  period  of  schooling  which  lacks  this  incen- 
tive must  be  a  barren  tract  in  the  experience  of  the  pupil 
who  has  traversed  its  dull  course. 

I  have  thus  here,  as  always,  striven  to  avoid  what  may 
be  deemed  the  most  insidious  and  mischievous  of  all  mis- 
takes in  teaching  and  textbook-making,  —  such  a  stooping 


INTRODUCTION.  17 

to  the  fancied  incapacities  of  pupils  as  requires  the  ob- 
scuration of  pure  thought,  the  blurring  and  distortion  of 
truth  by  substituted  analogies  and  illustrations.  Half- 
truths  are  dangerous.  Pupils  nurtured  on  such  philosoph- 
ical pap  too  often  take  up  the  role  of  teachers  without 
deepened  insight,  and  the  spawn  of  error  procreates  with 
the  fecundity  so  characteristic  of  parasites.  This  mis- 
take of  shutting  up  all  vistas  into  regions  not  presently 
under  exploration  stultifies  the  learner,  and  necessitates 
a  weary  process  of  unlearning  at  each  stadium.  It  is  in 
the  intellectual  sphere  the  analogon  of  that  contemptible 
principle  of  school  government  which,  in  the  sphere  of 
morals,  appeals  to  timidity  or  vanity,  and  depends  on 
espionage,  basely  ignorant  that  '^  the  human  character  is 
susceptible  of  other  incitements  to  correct  conduct  more 
worthy  of  employ  and  of  better  effect."  * 

In  conclusion,  the  difficulties,  practical  and  theoretical, 
of  the  central  problem  in  this  little  work  entitle  it  to 
be  judged  with  leniency.  It  is  submitted  to  my  classes 
and  to  fellow-teachers  for  such  uses  as  it  may  deserve. 

ARTHUR  LEFEVRE. 
University  of  Texas,  January,  1896. 


*  Thomas  Jefferson,  quoted  in  "  The  University  and  the  Common- 
wealth," an  address  by  Professor  Thornton  of  the  University  of  Virginia, 
delivered  in  the  University  of  Texas  on  Commencement  Day,  1894. 


SYLLABUS. 


NUMBER    AND    ITS    ALGEBRA. 

I.     Primary  Number. 

1.  Whole,  Litegral,  Natural,  Exact,  are  all  terms  in 
vogue  to  designate  the  primary  concept  of  number.  The 
former  two  are  equivalent,  and  objectionable  as  equally 
applicable  to  positive  and  negative  numbers.  They  thus 
fail  of  exact  designation.  Though  they  would  hardly  be 
chosen  de  novo,  they  may  be  retained  in  that  one  of  their 
present  uses  which  is  really  proper,  viz.,  to  designate  Pri- 
mary Numbers,  and  their  negatives  after  the  distinction  of 
positive  and  negative  has  been  clearly  made.  Each  of  the 
latter  two  is  repugnant  to  any  concept  of  number  adequate 
to  the  comprehension  of  mathematical  sciences.  No  num- 
ber must  be  conceived  as  either  unnatural  or  inexact.  The 
ratio  of  absolutely  incommensurable  sects,  the  diagonal  of 
a  square  and  its  side,  for  instance,  is  just  as  exactly  what 
it  is  as  the  ratio  of  an  inch  to  a  foot.  By  the  term  primary 
number  no  question  is  begged,  and  the  very  name  points  to 
the  development  so  soon  found  necessary. 

2.  Primary  Number,  a  normal  and  universal  creation  of 
the  human  mind,  applies  originally  only  to  artificial  wholes, 
discrete  aggregates.     The  group  whence  "  twelve "  is  ab- 

19 


20  NUIMBER   AND   ITS   ALGEBRA. 

stracted  must  be  conceived  as  a  Avhole  before  discriminated 
into  "  twelve."  Number  is  in  nowise  a  sense-perception  ; 
it  is  purely  the  product  of  a  rational  process.  Because  the 
adult  finds  a  number  concept  in  his  mind  when  a  group  of 
objects  is  attended  to,  he  must  by  no  means  suppose  any 
such  concept  in  the  mind  of  a  child,  though  the  same  objects 
be  attended  to.  The  objects  may  not  even  be  a  groiqj  at 
all  to  the  child.  Adults  forget  how  gradually  any  idea 
developed  in  their  minds.  Neither  is  the  concept  one 
necessarily  in  the  mind  of  a  child  when  a  single  object 
is  attended  to.  The  mind  of  the  child  is  inclined  to  be 
absorbed  in  sense  facts.  The  concept  one  is  only  in  con- 
trast to  the  concept  many.  It  is  not  my  purpose  to  inves- 
tigate the  origin  and  psychological  processes  of  these 
concepts,  one  and  many ;  nor  how  it  comes  to  pass  that 
the  infant  mind  slowly  tends  to  group,  aggregate,  make 
wholes  of,  distinct  individual  objects  of  sense-perception. 
It  is  enough  to  point  out  that  from  these  concepts  the 
primary  concept  of  number  springs.  Various  manys  are 
specialized,  and  so  distinct  numbers  arise  in  the  mind. 
We  will  not  enter  upon  the  question  of  infant  psychology 
concerning  the  stages  at  which  the  manys  are  specialized 
into  ''two,"  "three,"  etc.  It  may  be  remarked,  however, 
that  the  special  many  "  two  "  is  recognized  very  early  and 
long  before  "  three."  The  concepts  one,  many,  two,  come 
almost  together ;  and  then  after  a  long  gap  further  special- 
izations are  attained  —  another  distinct  gap  perhaps  coming 
after  fo^ir. 

3.  Definition-.  —  Primary  Number  is  an  abstraction 
from  a  group  of  objects  which  represents  their  individual 
existence. 

4.  Each  number-picture  of  a  group  is  wholly  abstract, 


COUNTING.  21 

in  that  it  represents  the  individual  existence  of  the  ele- 
ments of  the  group  and  nothing  more.  For  use  in  pictur- 
ing special  manys  a  system  of  abstract  elements  is  framed, 
where  no  characteristic  of  any  element  is  retained  beyond 
its  simple  separateness  from  all  others.*  This  brings  us  to 
Counting. 

II.    Counting. 

5.  The  fundamental  concept  of  primary  numbers  is 
prior  to,  prerequisite  for,  not  derived  from,  <'  Counting." 
The  word  is  used  in  two  senses,  though  its  general  syno- 
nyms, numeration  and  enumeration,  seem  sometimes  par- 
ticularly assigned  to  the  first  meaning.  In  the  first  sense. 
Counting  is  the  naming  of  primary  numbers.  This  nam- 
ing, if  carried  to  any  great  extent,  must  of  necessity  be 
methodical,  and  of  course  the  numbers  must  be  conceived 
before  named.  In  the  second  sense,  Counting  is  essentially 
the  numerical  identification,  by  a  one-to-one  correspond- 
ence, of  an  unfamiliar  with  a  familiar  group.  In  this 
meaning.  Counting  consists  in  assigning  to  each  individual 
in  a  group  one  distinct  individual  in  a  familiar  fixed  series 
of  different  things  —  originally  the  fingers,  usually  a  fixed 
series  of  different  words,  or  different  marks. 

6.  We  must  pass  by  many  interesting  facts  and  theories 
concerning  word- numerals  (i.e.,  fixed  series  of  different 
Avords  used  for  counting)  as  belonging  to  the  domain  of 
language,  only  remarking  that  etymology  confirms  what 
might  have  been  surmised,  that  the  fingers  were  the  origi- 
nal series  of  things  which  mankind  made  use  of  to  apply 
in  thought  to  a  group  of  objects  in  order  to  count  them. 


*  Vide,  Number,  Discrete  and  Continuous,  Halsted,  cliap.  i. 


22  DUMBER   AND   ITS   ALGEBRA. 

7.  lu  all  systems  of  numeration  or  counting  (in  the  first 
of  the  senses  defined  in  Section  5  )  it  soon  becomes  neces- 
sary, from  the  very  limitations  of  human  memory,  to  form 
or  mark  off  a  numerical  group  which  the  reckoner  can  peri- 
odically repeat.  Otherwise  there  would  be  no  end  to  the 
number  of  different  words  required.  The  number-group 
chosen  by  a  majority  of  races  at  a  pre-historic  time,  and 
for  the  reason  that  we  possess  ten  fingers,  is  ten.  As  soon 
as  any  such  group  has  been  chosen,  it  becomes  easy  to 
express  by  a  few  number-names  any  number  within  the 
mental  scope  of  the  speakers. 

For  instance,  in  English  with  fifteen  words  (two  of 
which  are  disfiguringly  superfluous)  and  two  significant 
suffixes,  i.e.,  with  seventeen  words,  any  number  whatever 
may  be  expressed. 

8.  The  student  should  write  out  a  detailed  explanation 
and  criticism  of  the  English  series  of  word-numerals,  not- 
ing the  superfluous  words,  also  such  as  are  not  internally 
suggestive  of  their  relation  to  the  fundamental  group,  and 
■^hat  larger  groups  lack  the  simple  name  that  would  be 
suggested  by  symmetry.  Let  him  then  compare  the  Eng- 
lish series  with  that  of  some  other  language,  noting  where 
the  English  is  better,  and  where  worse,  than  the  other. 
All  this  totally  irrespective  of  any  system  of  notation,  and 
purely  as  a  question  of  thought  and  language. 

According  to  the  best  practice  the  last  of  the  larger 
groups  to  receive  a  simple  name  for  repetition  is  the  mil- 
lion. Charles  W.  Merrifield,  F.E.S.,  observes,  "  It  is  worth 
while  to  remark  that  as  regards  billions  there  is  a  differ- 
ence between  the  French  and  English  practice ;  in  French 
a  hillion  (or  milliard)  is  one  thousand  millions,  in  English 
a  billion  is  a  million  millions,  .  .  .  the  word  is  seldom  used 


COUNTING .  23 

in  our  language.  .  .  .  The  old  books  use  a  scale  of  this 
kind:  A  million  of  millions  is  a  billion,  a  million  of 
billions  is  a  trillion,  and  so  forth;  but  these  names  are 
never  used  in  practice,  and  can  hardly  be  said  to  belong 
to  the  language  of  arithmetic  or  to  English  speech.''  *  In 
the  late  vulgar  use  of  the  Avord  in  American  newspapers, 
billion,  of  course,  signifies  one  thousand  millions ;  for  it  is 
a  comment  upon  the  vastness  of  such  numbers  that  even 
the  Fifty-first  Congress  could  not  expend  the  thousandth 
part  of  a  billion  dollars  in  the  sense  of  one  million 
millions. 

9.  By  the  method  just  discussed  a  distinct  name  is 
given  to  each  element  in  the  series  of  counters;  and  in 
counting  the  elements,  the  units,  the  ones  in  any  discrete 
magnitude  or  manifoldness,  a  one-to-one  application  is  made 
to  this  series  of  names  in  a  fixed  order.  The  order  being 
learned  by  rote,  any  word-numeral,  by  suggesting  its  defi- 
nite place  in  the  fixed  series  of  words,  recalls  all  those 
gone  before ;  and  from  this  comparison  the  mind  conveys 
or  receives  an  exact  notion  of  the  number  of  individuals 
in  the  group  of  objects  numerically  characterized  by  any 
such  number-name. 

10.  These  number-names  are  sometimes  called  cardinal 
numbers,  to  distinguish  them  from  a  series  called  ordinal 
numbers.  Ordinals  have  little  to  do  with  arithmetic,  the 
distinction  belonging  to  grammar  :  instead  of  saying  the 
last-counted  one  of  five  objects,  we  may  say  the  fifth,  etc. 
These  concepts,  first,  second,  third,  fourth,  etc.,  are  the 
"  ordinal  numbers." 

*  Arithmetic  and  Mensuration,  p.  4,  Longmans,  Green,  &  Co.,  1882. 


24  NUMBER   AND   ITS   ALGEBRA. 

III.    Primary  Number.  —  Some  Fundamental  Theory. 

11.  The  number  of  objects  in  one  group  is  said  to  be 
equal  to  the  number  in  another  when  their  units  being 
counted  (vide  §  5)  come  to  the  same  finger,  the  same  nu- 
meral-word or  mark.  That  is  to  say,  two  primary  numbers 
each  equal  to  a  third  are  equal.  Also,  of  two  such  num- 
bers one  is  always  less  than,  equal  to,  or  greater  than, 
the  other,  according  as  in  a  one-to-one  application  to  the 
counter-series  the  process  ends  with  a  prior,  the  same, 
or  a  subsequent,  element.  Also  a  primary  number  may  be 
added  to  itself  so  as  to  double.  (Cf.%  229.)  I  am  not 
concerned  whether  these  dicta  be  regarded  as  axioms  or  as 
postulates. 

Primary  number  is  thus  at  once  classed  as  a  magnitude. 

(Indeed  it  may  be  that  the  method  of  Hegel's  dialectic 
of  the  mathematical  catgeories  would  display  magnitude 
and  number  as  essentially  the  same ;  that  is,  as  co-ordinate 
transitions  to  the  same  ultimate.  At  least,  any  object  — 
as  a  line,  a  surface,  a  solid,  a  time,  a  temperature  —  is  a 
TYiagnitude  or  manifoldness  only  as  number,  in  the  final 
concept  thereof,  can  be  abstracted.) 

12.  There  is  much  of  prime  import  to  be  said  of  meas- 
urement {vide  Chapter  XIII.),  but  it  may  be  remarked  in 
this  connection  that  the  concept  of  measurement  develops 
pari  passu  with  that  of  number.  To  the  man  whose  con- 
cept of  number  is  only  what  has  been  defined  as  primary 
number,  measurement  is  hardly  to  be  distinguished  from 
counting.  Por  measurement  of  discrete  magnitude  is 
counting ;  and  to  the  intelligence  supposed  there  is  no  real 
measurement  of  continuous  magnitude,  but  any  continuous 
magnitude  is  "measured"  by  violently  discreting  it,  and 


NOTATION.  25 

counting  the  units  contained,  the  residue  being  regarded  as 
merely  a  fractional  redundancy.  In  short,  he  measures  in 
what  are  popularly  called  "  round  numbers."  True  meas- 
urement of  continuous  magnitude  is  conceivable  only  under 
the  developed  concept  of  number  which  includes  ratios. 

13.  Theorem.  ■ —  Primary  Number  is  independent  of  the 
order  of  counting. 

This  fact  is  discerned  immediately  from  the  individuality 
of  the  objects  in  the  group.  Since  in  counting  the  cor- 
respondence is  one-to-one,  the  same  extent  of  the  counter- 
series  is  always  necessary  and  sufficient  to  correspondence 
with  any  group  of  objects  in  whatever  order  they  be  applied 
to  the  counters. 

The  obviousness  of  this  truth  must  not  blind  to  its 
importance ;  for,  as  Clifford  affirms,  "  upon  this  fact  the 
whole  of  the  science  of  number  is  based."  * 

IV.    Notation. 

14.  Notation  is  primarily  the  representation  of  primary 
numbers  by  written  symbols ;  but  in  the  developed  science 
of  arithmetic  it  must  include  the  symbolic  representation 
of  ways  of  combining  numbers,  and  qualitative  distinc- 
tions, which  arise  upon  investigation.  Notation  in  the 
primary  sense  is  intimately  blended  with  numeration,  for 
it  is  merely  the  recording  of  the  results  of  counting.  It 
is  of  vast  importance,  however,  and  a  good  invention  for 
the  purpose  could  have  been  no  easy  task  ;  because  cen- 
turies on  centuries  passed  after  a  symmetrical  system  of 
numeration  had  been  developed  in  thought  and  language 

*  Common  Sense  of  the  Exact  Sciences,  W.  K.  Clifford,  chap.  i. 


26  NUMBER   AND   ITS   ALGEBRA. 

before  a  thoroughly  fitting  notation  was  achieved.  Whether 
the  beautifully  simple  and  perfect  algorithm  now  so  famil- 
iar to  little  children  was  perfected  at  a  single  stroke  of 
genius  on  the  part  of  a  nameless  Hindoo,  or  was  a  grad- 
ually consummated  invention,  history  does  not  reveal. 

15.  Just  as  we  passed  over  the  etymology  of  numeral 
words,  we  must  pretermit  interesting  facts  and  surmises  as 
to  how  each  written  sign  came  to  have  its  particular  mean- 
ing in  the  various  series  of  signs  which  mankind  has  in 
times  past  employed  or  still  uses.  Such  signs  for  number 
are  older  than  any  other  form  of  writing,  older  even  than 
the  development  of  language  in  the  denary  system.  For 
an  entertaining  monograph  on  this  subject,  consult  Profes- 
sor Eobertson  Smith's  article  on  "  Numerals  "  in  the  ninth 
edition  of  the  Encycloprndla  Britminica,  from  which  much 
of  the  following  section  has  been  taken. 

16.  The  simplest  representation  of  unity  is  a  single 
stroke.  The  next  step  would  be  to  devise  a  sign  to  repre- 
sent a  definite  group  of  strokes,  as  it  would  be  confusing 
to  repeat  single  strokes  too  often.  Soon  a  sign  for  a  defi- 
nite group  of  the  primary  groups  would  be  required.  The 
Babylonian  inscriptions  well  exemplify  this  simplest  mode 
of  notation.  The  mark  for  unity,  a  vertical  arrow-head,  is 
repeated  up  to  ten,  whose  symbol  is  a  barbed  sign  pointing 
to  the  left.  These  by  mere  repetition  serve  to  express 
primary  numbers  up  to  one  hundred,  for  which  a  new  sign 
was  employed. 

The  most  important  principle  of  meaning-signified-by- 
position  appears  in  this  system.  Though  the  symbol  of 
the  smaller  number  put  to  the  right  of  the  hundred  symbol 
represented  addition,  the  same  symbol  to  the  left  repre- 
sented a  multiplier.      This  principle  was  still  more  signifi- 


NOTATION.  27 

cant  in  another  system  developed  by  the  Babylonians. 
Strange  to  say,  they  oftened  reckoned  by  powers  of  sixty, 
calling  sixty  a  soss,  and  sixty  times  sixty  a  sar.  Survivals 
of  this  sexigesimal  method  remain  in  our  divisions  of  time, 
angles,  and  the  circle.  For  example,  the  square  of  59  is 
found  recorded  (translating  into  our  symbols)  58.1,  that  is, 
58  soss  and  one  (58  X  60  -(-  1) ;  but  on  the  same  tablets  the 
cube  of  30  is  recorded  7.30,  that  is,  7  sai^  and  30  soss.  We 
thus  see  that  because  they  had  devised  no  sign  for  zero,  it 
could  only  be  left  to  the  judgment  of  the  reader  whether 
sixty  or  its  square  was  intended. 

After  alphabets  became  established  in  a  fixed  order,  they 
began  to  lend  themselves  to  numerical  notation.  In  the 
old  Greek  notation,  said  to  go  back  to  the  time  of  Solon, 
and  often  called  the  Herodian  system,  after  Herodian  who 
described  it  in  a  work  written  about  200  a.d.,  1  stood  for 
one,  n  (-TrivTe)  for  five,  A  (Se'/ca)  for  ten,  H  (e/carov)  for  hun- 
dred, X  (xlXlol)  for  thousand,  M  (/xupt'ot)  for  ten  thousand. 
As  an  artifice  of  condensation  a  great  11  enclosing  any  sym- 
bol signified  five  times  the  number  represented  within. 
Another  application  of  alphabets  is  more  to  my  purpose. 
In  this  system  (common  to  Greeks,  Syrians,  and  Hebrews 
—  in  Greece  displacing  the  Herodian),  the  first  nine  letters 
stood  for  units,  the  second  nine  for  tens,  the  third  nine  for 
hundreds,  and  diacritic  marks  below  the  first  nine  trans- 
formed them  into  thousands.  A  great  M  multiplied  the 
number  after  whose  sign  it  was  written  by  ten  thousand. 
The  notation  was  subsequently  improved  by  writing  the 
greater  element  always  to  the  left,  thus  disjieyising  with  the 
diacritic  marks.  The  regular  alphabet  furnishing  only 
twenty-four  letters,  the  necessary  twenty-seven  were  made 
up  by  calling  in  two  old  letters  no  longer  used  in  phonetic 


28  KUMBEE,   AND   ITS   ALGEBRA. 

writing,  to  signify  six  ancV  ninety,  and  a  final  symbol  called 
samjn  represented  nine  hundred.  Approaches  still  nearer 
to  our  algorithm  were  devised  by  Greek  mathematicians, 
notably  Archimedes  and  ApoUonius  of  Perga;  but  in  all 
the  lack  of  the  zero  rendered  the  systems  very  imperfectly 
adapted  to  calculation,  however  perspicuous  as  a  record. 

Only  one  more  system  can  be  glanced  at  before  survey- 
ing our  own.  This,  known  as  the  Roman,  we  are  still 
familiar  with.  It  more  resembles  the  clumsy  Herodian 
than  the  later  Greek  notation.  The  symbols  were,  1  =  1, 
V  =  5,  X  =  10,  L  =  50,  C  =  100,  D  =  500,  M  =  1,000. 
Some  older  forms  were  afterwards  discarded.  To  the  ex- 
tent of  a  few  subtractive  forms  (IV  =  4,  IX  =  9,  XL  =  40, 
XC  =  90,  and  occasionally  IIX  =  8,  XXC  =  80)  some  mean- 
ing is  attached  to  position,  but  in  a  way  rather  to  hinder 
calculation. 

In  a  mechanical  contrivance,  used  in  Europe  from  a  very 
early  date,  was  attained  the  nearest  approach  to  our  own 
system.  The  abacus  (which  could  be  ruled  on  waxen  tab- 
lets or  roughly  drawn  on  the  ground),  in  a  permanent  form, 
consisted  of  a  frame  in  which  by  one  means  or  another  sets 
of  counters  were  kept  in  separate  rows  or  columns.  These 
columns  might  represent  various  denominations  of  money 
value,  or  weight,  or  units,  tens,  hundreds,  thousands,  etc. 
In  the  latter  case  there  should  be  only  nine  counters  in  a 
column.  From  such  an  abacus  there  are  but  two  steps  to 
our  notation:  first,  to  establish  marks  to  represent  respec- 
tively one,  two,  ...  or  nine  counters  in  any  column ;  sec- 
ond, to  conceive  a  sign  for  a  vacant  column.  The  inven- 
tion of  our  nine  digits  and  zero  came  slowly.  The  history 
is  very  obscure.  Our  "  Arabic  "  system  is  of  Indian  origin, 
but  appears  to  have  been  introduced  into  Europe  by  the 


NOTATION.  29 

Arabs.  It  has  been  traced  as  far  back  as  the  fifth  century 
of  the  Christian  era  in  India,  but  does  not  seem  then  to 
have  been  a  novelty.  Hindoo  writers  nowhere  lay  claim  to 
its  invention.  It  was  probably  brought  to  Baghdad  in  the 
eighth  century.  In  the  ninth  century  Abu  Jafar  Moham- 
med al-Ivharismi  published  a  work  on  the  subject,  and  by 
the  tenth  it  had  spread  into  general  use  throughout  the 
Arabian  world.  About  the  twelfth  century  it  began  to  be 
received  by  Christian  Europe.  Arithmetic  using  this  sys- 
tem was  called  by  the  barbarous  name  AJgoritmus  (our 
algorithm),  probably  a  derivation  from  al-Kharizmi.  Leo- 
nardo of  Pisa  promulgated  the  matter  in  the  West,  and 
Maximus  Planudes  in  the  East.  The  word  zero  is  perhaps 
derived  from  the  Arabic  sifr,  through  zepliyro,  used  by 
Leonardo.  The  algoritmus  was  at  first  used  chiefly  in 
astronoDiical  tables,  etc.  (e.g.,  those  published  about  1252 
by  Alfonso  the  Wise).  Gradually  the  immense  superior- 
ity of  the  system  above  all  others  became  apparent,  and  it 
has  long  been  used  by  all  civilized  nations.  In  winning  its 
way  there  was  some  confusion  with  prevailing  notations  : 
e.g.,  such  forms  as  X2  =  12  and  504  =  54  are  found,  where 
the  very  essence  of  the  method  is  lost  sight  of. 

17.  Our  notation  exactly  conforms  to  our  system  of  nu- 
meration. The  symmetr}"  or  regularity  of  the  notation,  how- 
ever, is  perfect.  No  such  anomaly  as  is  found  in  the  word 
"  eleven  "  or  "  twelve  "  is  tolerated. 

The  familiar  symbols  always  mean  one,  two,  ...  or 
nine ;  but  they  signify  units,  tens,  ten-tens,  ten  ten-tens, 
etc.,  according  to  their  position  in  the  first,  second,  third, 
fourth,  etc.,  place,  counting  from  right  to  left.  That  is  to 
say,  they  represent  in  definite  positions  corresponding 
powers  of  ten. 


30  NUMBER    AND   ITS   ALGEBRA. 

Under  the  generalization  of  "  powers  "  the  notation  at 
once  lends  itself  to  the  expression  of  fractions,  the  expo- 
nents becoming  negative.    For  example,  4072.605  means  — 

(10)3  -^  4  ^  (10)2  X  0  +  (10)^  X  7  +  (10)°  X  2  +  (10)-^ 
X  6  +  (10)--  X  0  +  (10)-^  X  5. 

It  is  in  this  regular  use  of  a  base-number  that  the  merit 
of  the  system  consists,  and  by  no  means  in  the  choice  of 
the  base  ten.  Our  decimal  system  is  a  perfect  instrument, 
exciting  the  grateful  admiration  of  every  enlightened  stu- 
dent of  science,  not  because  it  is  decimal,  but  because  the 
digit  figures  by  means  of  the  zero  always  express,  in  their 
orderly  position,  to  left  or  right  of  a  point,  ascending  ot 
descending  powers  of  one  basal  number. 

In  regard  to  the  particular  base,  ten,  it  may  be  remarked 
that,  while  it  were  idle  to  think  of  changing  the  confirmed 
habits  of  language,  it  is  clear  that  ten  is  an  inconvenient 
base.  Twelve  would  be  better.  To  see  this,  it  is  enough 
to  express  decimally  and  duodecimally  a  few  simple  frac- 
tions. '  ,  t '"' 

Decimally  1/3  =  0.3333333  .  .  .  Duodecimally  1/3  =  0.4 
1/4  =  0.25  1/4  =  0.3 

1/6  =  0.1666666  ...  1/6  =  0.2 

1/8  =  0.125  1/8  =  0.16 

1/9  =  0.1111111  ...  1/9  =  0.14 

18.  A  thoughtful  consideration  of  our  notation  will 
enable  the  student  to  adapt  its  essential  principles  to  any 
base.  So  long  as  he  feels  hesitation  in  doing  this,  he 
may  be  sure  he  does  7wt  understand  what  he  has  deemed 
so  familiar. 


ALGEBRA.  31 

It  is  obvious  that  a  number  is  the  same,  whether  ex- 
pressed in  tens,  or  dozens,  or  scores,  —  to  take,  for  example, 
two  numbers,  other  than  ten,  familiar  as  bases  to  English 
minds,  but  which  have  never  been  developed  into  symmet- 
rical systems  of  counting. 

Plainly  the  number  of  digit  figures  required  is  one  less 
than  the  base ;  since  10  must  represent  the  base,  whatever 
it  may  be. 

19.  The  student  should  express  various  numbers,  inte- 
gral and  fractional,  on  various  bases,  employing,  say,  the 
letters  of  the  English  alphabet  in  order,  for  additional 
digit  figures  when  more  than  nine  are  required.  He 
should  also  perform  additions,  subtractions,  multiplica- 
tions, divisions,  with  numbers  so  expressed. 

There  is  no  other  way  to  test  or  gain  a  thorough  com- 
prehension of  the  notation  so  glibly  used.  Such  an  exer- 
cise will  remedy  many  defects,  and  will  be  found  to  repay 
amply  the  slight  cost  in  time  and  labor.     (  Vide  §  277  et  seq.) 

V.    Algebra. 

20.  An  algebra  is  an  artificial  language.  Its  symbols 
have  laws  of  combination ;  but  these  laws  are  the  expres- 
sion of  actual  properties  and  relations  of  the  subject-matter, 
not  laws  of  the  algebra  in  any  immediate  sense.  There  is 
no  such  thing  as  an  algebraic  law ;  there  are  only  alge- 
braic conventions.  The  peculiar  advantage  of  an  algebra 
is  that  actual  relations  are  given  manifestations  which  can 
be  experimented  upon,  according  to  organized  processes,  to 
give  new  knowledge.  The  first  algebra  was  slowly  formed 
through  centuries  to  investigate  the  properties  of  number.* 

*  Vide  Halsted's  Number,  Discrete  and  Continuous,  §  1. 


32  NUMBER   AND   ITS   ALGEBRA. 

21.  Algebras  of  formal  logic,  of  physics,  of  geometry, 
have  been  developed  to  more  or  less  usefulness.  In  these 
the  subject-matters  immediately  discoursed  of  are  respec- 
tively logical,  physical,  geometrical  entities  and  their  rela- 
tions and  combinations ;  just  as  the  algebra  with  which 
we  have  to  do  discourses  of  number  and  its  relations  and 
combinations.  The  specific  entities,  relations,  and  combi- 
nations in  any  case  require  definition,  and  yield  laws  upon 
their  own  merits. 

22.  A  geometrical  algebra  must  be  clearly  distinguished 
from  that  supremely  powerful  and  distinctively  modern 
branch  of  analysis  (from  whose  establishment  by  Des- 
cartes, 1637,  dates  the  modern,  the  scientific  era)  commonly 
called  Analytical  Geometry.  In  this  discipline  the  alge- 
bra is  still  of  number,  the  immediate  subject  of  discourse 
is  ever  number ;  but  under  systematic  conventions  the 
algebra  talks  in  numbers  about  geometry,  just  as  it  might 
be  made  to  talk  about  money  or  temperatures.  In  a  true 
and  proper  algebra  of  geometry,  a  and  b  might  represent 
sects,*  and  ab  be  defined  as  the  definite  plane  surface 
known  as  the  rectangle  of  a  and  b.  In  this  case  there 
could  be  no  ratio  between  ab  and  a.  Also  a^  would  mean 
the  actual  surface,  the  square  on  a ;  a^,  the  actual  solid, 
the  cube  on  a ;  and  a^,  etc.,  would  be  devoid  of  meaning  in 
tri-dimensional  space. 

23.  However  mechanically  we  may  at  times  use  the 
symbols,  it  cannot  be  too  much  emphasized  that  in  the 
algebra  of  number  each  expression  must  be  a  rational  dis- 
course upon  number  to  any  mind,  or  to  that  mind  it  is 
nonsense,  or  rather  a  blank,  like  a  sentence  in  an  unknown 


*  Definite  pieces  of  straight  lines. 


ALGEBRA.  33 

tongue.  Clifford  maintains,  "  We  may  always  depend  upon 
it,  that  algebra  which  cannot  be  translated  into  good  Eng- 
lish and  sound  common  sense,  is  bad  algebra."  * 

24.  Although  of  immense  utility,  the  algebra  of  number 
must  not  be  conceived  as  theoretically  necessary  to  the 
investigations  it  has  so  signally  served.  The  instrument 
has  been  practically  prerequisite  to  the  results  that  have 
been  attained  on  account  of  the  limitations  of  mankind's 
power  of  attention  to  complex  details  without  symbolic 
expression,  but  its  essentially  derivative  nature  must  not 
be  lost  sight  of.  Under  any  concept  subversive  of  this 
relation — the  fallacy  being  even  more  baneful  Avhen  im- 
plied than  when  explicit  —  the  study  of  an  algebra  be- 
comes abusive  of  the  noblest  qualities  of  mind ;  and  no 
irrational  skill  in  the  use  of  the  tool  can  compensate  for 
the  intellectual  debasement  which  is  the  price  of  content- 
ment in  its  use  and  study  upon  such  terms.  It  is  as  if 
one  conceived  the  vocabulary  of  a  spoken  language  as 
independent  of  the  constructive  thought ;  back  of  any 
mode  of  symbolic  expression  must  lie  the  substantial 
thought. 

To  understand  our  algebra  of  number,  we  must  under- 
stand number.  However  difficult  the  task,  it  cannot  hon- 
estly be  shirked. 

25.  Many  eminent  mathematicians,  to  say  nothing  of 
popular  text-books,  persist  in  seeking  explanation  of  the 
algebra  of  number  in  the  facts  of  geometry.  They  seem 
blind  to  the  view  that  it  is  only  adaptations  of  number 
that  they  thus  discover;  that  it  is  numbers,  not  lines, 
surfaces,  solids,  that  they  deal  Avith,  even  when  they  so 


*  Common  Sense  of  the  Exact  Sciences,  p.  21. 


34  NUMBER   AND   ITS   ALGEBRA. 

usefully  make  the  algebra  of  number  "talk  geometry."* 
The  individual  symbol  in  trigonometry  or  analytical  geom- 
etry, for  examjjle,  never  means  the  geometrical  concept. 
An  equation  may  under  a  proper  system  of  interpretation 
describe  a  line  ;  but  no  x  or  y  in  it  ever  means  a  line, 
but  the  length  of  a  line,  which  is  a  ratio,  a  number. 
Would  it  be  less  sophisticated  to  try  to  discover  the  nature 
and  properties  of  number  by  studying  temperatures,  be- 
cause, forsooth,  an  algebraic  equation  may  under  appro- 
priate conditions  talk  temperatures  as  well  as  geometry  ? 
The  confusion  arising  from  such  misconceptions  is  well 
exemplified  in  the  following  quotation  from  an  essay  by 
E.  W.  Hyde  in  the  American  Journal  of  Mathematics  for 
September,  1883,  p.  3  :  — 

'<  If,  in  the  equation  1/1  =  1x1,  1  be  taken  as  a  unit 
of  length,  then  the  meinbers  of  the  equation  have  evi- 
dently not  the  same  meaning,  1/1  being  merely  a  numeri- 
cal quantity,  while  1x1  is  a  uuit  of  area  ;  it  being  a 
fundamental  geometric  conception  that  the  product  of  a 
length  by  a  length  is  an  area,  that  of  a  length  by  an  area 
a  volume,  while  the  ratio  of  two  quantities  of  the  same 
order  as  that  of  a  length  to  a  length  is  a  mere  number 
of  the  order  zero." 

So  far  from  just  are  these  observations  that  one  would 
suppose  it  clear  to  any  student  of  the  subject  that  the 
physical  fact  is  the  line,  the  surface,  the  solid,  and  that 
the  length,  the  area,  the  volume,  are  numbers,  viz.,  the 
ratios  of  the  line,  surface,  and  solid  respectively  to  other 
magnitudes  of  like  kind  chosen  arbitrarily  as  units.  It  is 
a  theorem  which  we  have  established  geometrically,  that 

*  A  felicitous  i)hrase  of  Dr.  Halsted's. 


ALGEBRA.  35 

the  ratio  of  the  rectangle  of  two  sects  to  the  square  on 
any  third  sect  equals  the  product  of  the  ratios  of  the  two 
given  sects  to  the  third  sect.  That  is  to  say,  the  area  of 
a  rectangle  equals  the  product  of  the  lengths  of  two  adja- 
cent sides,  it  being  distinctly  understood  that  the  unit- 
surface  is  the  square  on  the  unit-line.  This  truth  having 
been  established,  consistent  numerical  statements  may  be 
referred  to  such  spatial  entities.  It  is  only  and  always  in 
some  such  Avay  that  the  algebra  of  number  ''  talks  ge- 
ometry." 

26.  Objections  to  the  mistake  of  explaining  number  geo- 
metrically are  often  made  at  a  fatally  late  stage.  There 
are  writers  who  protest  against  geometric  definitions  of 
the  so-called  imaginary  numbers  after  having  supinely 
ignored  a  geometric,  or  some  unnumerical,  definition  of  —1. 
Their  alertness  comes  too  late  when  they  refuse  a  like 
definition  of  V—  1.  In  this,  as  in  many  other  cases,  it 
is  the  first  principles  that  have  been  neglected.  It  is 
futile  to  begin  inquiry  with  V  —  1.  AYith  beclouded 
concepts  of  prior  phases  of  number,  how  can  it  be  any- 
thing but  vain  to  attempt  to  be  critical  at  the  final  stage 
of  that  development  of  number  which  has  forced  itself 
alike  upon  the  most  practical  and  the  most  theoretical  ? 
{Cf.  §  192.) 

27.  Concerning  other  extant  or  possible  algebras  than 
that  of  number,  I  Avill  only  add  that  I  have  grave  doubts 
of  the  propriety  of  Professor  Macfarlane's  aspirations 
towards  a  final  and  comprehensive  algebra,*  ''  which  will 
apply  directly  to  physical  quantities,  will  include  and  unify 

*  "Principles  of  the  Algebra  of  Physics,"  by  A.  Macfarlane,  M.A., 
D.Sc,  LL.D.,  in  Proceedings  of  the  American  Association  for  the  Ad- 
vancement of  Science,  vol.  xl.,  18U1,  p.  (i5.    The  italics  are  mine. 


36  NUMBER    AND   ITS   ALGEBRA. 

the  several  branches  of  analysis,  and  when  specialized  will 
become  ordinary  algebra."  Far  from  being  the  "  special- 
ized" form,  the  algebra  of  number  appears  to  me  to  be 
the  very  generalization  sought  by  Dr.  ]\Iacfarlane ;  and  it 
is  algebras  of  physics,  vector  algebras,  etc.,  which  are  the 
specializations.  Number  itself  in  its  full  development 
appears  to  my  mind  the  very  ultimate  common  property 
of  all  quantity,  magnitude,  manifoldness  (^vlde  §  229)  what- 
soever. Search  for  further  generalization  seems  mistaken. 
I  set  forth  this  opinion  tentatively  and  in  all  modesty, 
fully  recognizing  Dr.  Macfarlane's  profound  learning  and 
skill  in  mathematics. 

Negative,  neomonic,  and  complex  numbers  afford  the 
qualitative  distinctions  under  Avhich,  it  seems,  the  algebra 
of  number  might  be  made  to  talk  physics  and  geometry 
to  our  full  satisfaction.  Should  it  be  found  inadequate  to 
the  needs  of  the  physicist,  of  course,  a  true  and  proper 
algebra  of  physics  may  be  fashioned.  The  physicists 
must  decide  this  question.  Might  not  better  results,  how- 
ever, be  attained  by  seeking  perfectly  satisfactory  means 
for  interpreting,  physically  or  geometrically,  numerical 
statements,  the  algebra  for  which  is  ready  to  hand,  than 
by  attempting  to  construct  any  real  algebra  of  physics  to 
"apply  directly  to  physical  quantities  "  ? 

I  may  invoke  here  the  authority  of  no  less  a  physicist 
than  James  Clerk  Maxwell.  After  pointing  out  the  con- 
tradictions which  would  otherwise  occur  in  calculation, 
he  says  :  "  We  shall  therefore  consider  all  the  symbols 
as  mere  numerical  quantities,  and  therefore  subject  to 
all  the  operations  of  arithmetic.  But  in  the  original 
equations  and  the  final  equations  in  which  every  term 
has  to  be  interpreted   in  a  physical  sense,  we  must  con- 


CALCULATION.  87 

vert  every  numerical  expression  into  a  concrete  quantity 
b}'  multiplying  it  by  the  unit  of  that  kind  of  quantity.'' 

28.  If  ^ye  will  regard  the  algebra  of  number  from  the 
standpoint  of  recognition  of  its  true  nature,  we  may  take 
up  its  natural  use  without  more  ado,  (Vide  §  156.)  There 
is  nothing  mysterious  about  the  algebraic  vocabulary,  or 
even  recondite  in  the  algebra,  liutil  we  reach  more  ad- 
vanced investigations  concerning  algebraic  form.  The 
original  obscurities  and  difficulties  are  in  the  arithmetic ; 
that  is,  in  the  theory  and  import  of  number  itself.  For 
the  most  j^art  I  shall  consider  what  is  algebraical  already 
familiar,  and  bend  all  energy  to  expounding  the  numerical 
content,  as  distinguished  from  the  algebraic  form.  (But 
see  §  236.) 

VI.      CALCUIiATIOX. 

29.  Calculation,  or  computation,  is  primarily  counting. 
As  its  methods  gradually  become  organized,  it  involves  a 
thorough  investigation  of  the  laws  of  thought,  which,  upon 
consideration  by  any  normal  mind,  will  be  seen  to  govern 
the  various  possible  combinations  of  numbers  and  the 
processes  of  these  combinations. 

30.  In  solving  particular  problems,  whether  concerning 
numbers  or  the  application  of  numbers  to  concrete  magni- 
tudes, it  is  to  be  borne  constantly  in  mind  that  all  that 
can  be  tauglit  in  general  terms  is  how  to  conceive  and 
perform  numerical  operations  ;  that,  knowing  this,  all  that 
remains  is  to  understand  the  terms  of  a  particular  problem 
and  the  properties,  real  or  conventional,  of  these  terms, 
under  which  they  yield  numerical  relations  ;  and  that  until 
one  recognizes  this  fact  he  cannot  take  the  first  rational 
step. 


38  NUMBER    AND   ITS   ALGEBRA. 

Although  it  is  proper  and  necessary  in  teaching  to  make 
constant  applications  of  pure  or  theoretical  arithmetic, 
yet  the  way  in  which  these  applications  are  presented 
in  ordinary  text-books  is  grossly  misleading.  There  is 
no  arithmetical  distinction  whatever  between  such  topics 
as  "  percentage,"  "  interest,"  "  discount,"  "  commission," 
'•'  brokerage,"  "  partial  payments,"  etc. ;  yet  from  flaring 
chapter-headings  the  distinctions  appear  co-ordinate  with 
those  between  numeration,  numerical  operations,  and  gen- 
eral devices,  such  as  methods  for  finding  the  greatest  com- 
mon submultiple,  or  the  least  common  multiple.  No  new 
arithmetical  lore  is  required  in  order  to  calculate  about 
these  mercantile  transactions ;  the  task  for  the  pupil  is 
merely  to  comprehend  a  few  technical  terms,  and  the 
numerical  relations  subsisting,  or  in  practice  assumed  to 
subsist,  among  them.  Nevertheless,  it  is  a  common  result 
of  the  misconceived  method  of  presentation  that  a  pupil 
fancies  he  is  advancing  to  a  new  development  of  the  arith- 
metic when  he  passes  from  ''commission"  to  "brokerage," 
for  instance,  as  if  there  were  the  faintest  arithmetical  dis- 
tinction between  calculating  a  percentage  on  the  value  of  a 
barrel  of  apples  and  the  value  of  a  block  of  capital  stock. 
In  like  manner,  pupils  often  make  pathetic  attempts  to 
excogitate  the  conventional  method  of  calculating  a  bal- 
ance due  on  an  account  Avith  partial  payments,  being 
blinded  by  incompetent  teaching  to  the  fact  that  the  data 
do  not  afford  numerical  relations  sufficient  to  a  definite 
theoretical  solution.  In  this  matter  (as  in  many  others 
in  Applied  Arithmetic),  arbitrary  convention  is  neces- 
sary to  a  solution ;  the  "  rule "  varies  with  the  practice 
of  individuals,  enactments  of  legislatures,  and  rulings  of 
courts  of  law  and  equity.     No  act  of  pope  or  parliament 


ARITHMETIC,  PURE  AND   APPLIED.  89 

could  affect  the  proper  decision  of  any  truly  arithmetical 
question.  Of  course  there  is  more  or  less  numerical  pro- 
priety in  the  substantial  justice  between  man  and  man 
which  is  sought  in  each  of  the  various  rules  for  calculating 
a  balance  after  partial  payments,  but  there  are  questions 
involved  whose  decision  is  not  afforded  by  inherent  numer- 
ical relations  of  the  facts. 

31.  In  teaching,  it  is  supremely  helpful  always  to  empha- 
size the  difference  between  Pure  Arithmetic  and  Applied 
Arithmetic.  The  pupil's  knowledge  is  surely  in  confusion 
unless  he  sees  the  fundamental  difference  between,  for  ex- 
ample, studying  how  to  find  a  least  common  multiple  (a 
matter  of  insight),  and  a  broker's  commission  (a  matter  of 
empiric  information  so  far  as  it  is  anything  new  when  met 
in  a  systematic  course).  Applied  Arithmetic  should  be 
presented  in  text-books  as  merely  selected  specimens  of 
many  other  practical  applications  which  could  be  made, 
and  as  problems,  not  as  new  arithmetical  topics.  The 
necessary  information  should  be  set  forth  in  an  entirely 
different  tone  from  that  in  which  arithmetical  matters 
proper  are  expounded.  The  particular  applications  usually 
made  are  sufficiently  well  chosen;  viz.,  calculations  concern- 
ing lines,  surfaces,  solids,  times,  weights,  temperatures,  and 
money  values,  Avith  special  reference  to  the  transactions  of 
mercantile  and  banking  business. 

But  no  candid  criticism,  even  the  most  cursory,  could 
avoid  complaint  on  account  of  the  usual  results  of  teaching 
the  metric  system  of  units.  Text-books  are  at  fault  here, 
rather  negatively  than  positively ;  though  some  are  found 
to  write  ImSdmScm,  when  the  system  was  devised  ex- 
pressly to  avoid  this —  it  is  as  if  one  should  write  1  dollar, 
3  dimes,  8  cents.     They  might  be  expected,  however,  to 


40  NUMBER    AND   ITS    ALGEBRA. 

put  tlie  matter  in  a]i  appreciative  and  tonic  way,  instead  of 
leaving  the  discovery  of  its  perfections  to  the  chance  alert- 
ness of  the  pupil.  Eor,  because  a  matter  is  perfectly  clear 
and  simple,  it  does  not  follow  that  it  will  be  so  esteemed. 
The  case  in  question  demonstrates  this  paradox.  The 
metric  system  of  units  was  invented  for  its  perfect  simpli-' 
city ;  yet,  pitiful  to  say,  it  remains  a  bugbear  to  the  average 
teacher  and  pupil  in  the  common  schools.  Nothing  could 
be  more  blind  and  irrational  —  it  is  exactly  as  if  an  Eng- 
lishman could  not  be  made  to  see  that  decimal  money  units 
are  simpler  for  all  calculation  than  pounds,  shillings,  and 
pence.  Any  student  may  be  sure  that,  unless  he  regards 
the  metric  system  as  perfectly  clear,  and  vastly  easier  than 
our  barbarous  English  units,  he  has  entirely  failed  to  un- 
derstand it  —  nor  could  one  fail  to  appreciate  it  who  really 
understood  anything  of  arithmetic.  Its  essential  merit  is 
twofold :  it  is  decimal,  and  therefore  fits  our  numerical 
notation ;  its  units  for  lines,  surfaces,  solids,  masses,  and 
weights  are  all  symmetrically  dependent  on  one  unit,  the 
linear. 

The  advantages  of  the  second  property  ought  to  be  as 
manifest  as  those  of  the  first ;  but  I  Avill  briefly  illustrate. 
If  the  volume  of  some  homogeneous  material  is  given  as 
2.76  cu.  m.  and  its  sp.  gr.  3.5,  the  weight  may  be  found 
by  multiplying  the  numbers  :  2.76  X  3.5  =  9.66  tonneaux, 
or,  pointing  oif  to  reduce  to  kilograms,  9660  kg. 

Now,  in  comparison  let  the  student  calculate  the  weight, 
given  volume  2  cu.  yd.,  7  cu.  ft.,  6  cu.  in.,  and  sp.  gr.  3.5. 
In  the  first  place,  exact  calculation  is  impossible  in  the 
English  units ;  for  the  pound  and  the  weight  of  a  cu.  yd.  of 
water  are,  of  course,  incommensurable.  The  first  task  is 
to  look  up  in  some  compendium  of  useful  information  the 


ARITHMETIC   AND   ALGEBRA.  41 

approximate  weight  of  a  cubic  yard  of  standard  water,  or 
of  a  cu.  ft.,  or  of  a  cu.  in.,  as  may  be  vouchsafed.  Then 
reduce  the  volumetric  terms  accordingly,  then  multiply, 
and  finally  (to  be  thoroughly  English)  reduce  the  approxi- 
mate decimal  fraction  of  the  pound  to  ounces  and  grains. 
One  who  has  stupidly  despised  the  metric  system  ought 
to  perform,  as  a  penance,  this  calculation  a  V Anglaise. 

32.  It  may  be  helpful  to  state  explicitly  that  I  always 
use  the  term  arithmetic  in  the  sense  of  the  Science  of 
Number. 

There  is  no  difference  either  in  subject-matter  or  in 
scope  between  arithmetic  and  the  algebra  of  number.  The 
distinction  made  by  the  term  algebra  refers  to  the  mode  of 
expression  (vide  §  20),  and  in  a  special  sense  to  the  pro- 
fou.ndly  important  subject  of  algebraic  forms  of  numerical 
expression.  Any  arithmetical  statement  is  of  particular 
numbers ;  while,  from  the  very  nature  of  the  conventions, 
algebraic  statements  are  general.  It  was  to  this  end  that 
algebra  was  invented  ;  but  it  must  never  be  forgotten  that 
any  algebraic  expression  may  be  made  particular  (x'ide 
§  23),  and  that  the  form  then  becomes  arithmetical. 

'^  Arithmetic  "  is  too  often  limited  (very  illogically  and 
contrary  to  the  best  practice)  to  denote  merely  some  primi- 
tive developments  of  the  science  of  number.  Even  the 
distinction  positive  and  negative  is  often  expressly  set 
forth  as  peculiarly  a  matter  of  algebra.  In  our  view  this, 
of  course,  is  utterly  subversive. 

Arithmetic  needs  and  uses  the  same  symbols  of  opera- 
tion and  qualitative  distinctions  a.s  the  algebra  of  number  ; 
indeed,  logically,  the  statement  should  be  made  the  other 
way,  viz.,  the  algebra  iTses  the  same  symbols  of  operation 
and  quality  as  arithmetic.     The   symbols  +,  —  (in  both 


42  NUMBER   AND   ITS   ALGEBRA. 

senses  of  each),  tlie  exponential  notation,  etc.,  belong 
equally  to  the  notation  of  arithmetic  and  to  number's 
algebra. 

Newton  preferred  to  call  the  algebra  Universal  Arith- 
metic. 

33.  It  would  be  a  very  good  exercise  for  the  student 
(especially  those  who  are  taking  this  course  in  order  to 
qualify  as  teachers  in  the  public  schools)  to  critically 
examine  some  text-book  on  arithmetic  which  he  has  heard 
extolled.* 

VII.     Primary  Number,  —  Numerical  Operations. 

34.  There  are  seven  distinct  numerical  operations. 
Three  of  these  are  direct,  and  four  inverses  of  these  three. 
The  three  direct  arise  from  three  different  modes  of  com- 
bining two  numbers,  and  the  four  inverse,  from  inverse 
problems,  viz.,  given  one  of  the  two  numbers  in  the  former 
combination  and  the  resulting  number,  to  find  the  second 
of  the  two  originals.  Inasmuch  as  two  of  the  direct  oper- 
ations are  commutative  (vide  §  38),  they  give  rise  each  to 
only  one  inverse ;  that  is,  it  is  the  same  problem,  given 
either  and  the  result,  to  find  the  other.  But  the  third  of 
the  direct  operations  is  not  commutative ;  and  it  therefore 
gives  rise  to  two  inverses,  it  being  a  very  different  prob- 
lem, whether  the  first  or  the  second  of  the  originals  be 
given,  to  find  the  other. 

These  seven  operations  are,  by  name,  Addition  and  its 
inverse,  Subtraction  ;  Multiplication  and  its  inverse,  Divis- 

*  Upon  a  reperusal,  this  expression  seems  almost  satirical,  since  it 
would  be  impossible  to  find  one  which  has  not  been  extolled.  I  let  the 
phrase  stand,  however,  in  all  its  innocent  irony. 


PRIMARY   NUMBER. — NUMERICAL   OPERATIONS.        43 

ion ;     Involution    and    its    two    inverses,    Evolution    and 
Finding  the  Logarithm. 

No  numerical  operations  have  been  developed  showing 
characteristics  essentially  different  from  these  modes  of 
operational  combination.  (For  fuller  discussion  of  this 
point,  vide  §  104.) 

35.  It  would  be  a  sad  comment  on  previous  instruction  if 
any  one  is  surprised  to  hear  of  the  seventh  of  these  funda- 
mental operations  ;  for  when  we  find  a^  =  c,  the  two  in- 
verse problems  are  equally  obvious ;  we  may  have  given 
b  and  c,  to  find  a  (this  is  familiar  as  evolution),  or  we 
may  have  a  and  c  given,  to  find  h.  Scientific  mathematicians 
(Cf.  Introduction,  p.  5)  are  to  blame  that  no  single  name 
denoting^  this  operation  is  current.  We  must  iise  the 
accepted  phrase,  finding  the  logarithm.  Because  the  pro- 
cess of  this  last  operation  is  comparatively  recondite,  is 
no  excuse  for  not  calling  attention  to  the  problem  in  the 
very  beginning  of  any  systematic  teaching  of  arithmetic. 
Indeed,  under  any  rational  instruction  it^  existence  could 
not  be  concealed.  It  should  be  said  to  the  pupil,  "  Evidently 
such  a  result  as  «*  =  c  presents  two  inverse  problems.  At 
this  stage  you  will  investigate  only  how  a  few  very  simple 
roots  may  be  extracted.  The  question  how  the  exponent 
or  logarithm  may  be  determined  must  be  deferred  until 
you  have  acquired  more  knowledge  and  greater  skill." 

36.  In  this  chapter  the  fundamental  operations  will  be 
tentatively  considered  for  primary  number.  It  will  be- 
come apparent  that  for  any  two  primary  numbers  the 
direct  operations  are  always  possible,  but  that  the  inverse 
operations  have  meaning  only  in  particular  cases.  Equally 
obvious  will  become  the  urgent  need  and  propriety  ot 
extending  the  concept  of  number,  both  for  the  theoretical 


44  NUMBER   AND   ITS   ALGEBRA. 

science,  and  its  application  to  the  measurement  of  concrete 
magnitudes, 

Tlte  numerical  symhols  of  the  algebra  evvployed  in  this 
chapter  are  general  only  for  irriniary  number ;  -j-  and  — 
have  only  their  operational  meanings.  This  strict  limita- 
tion must  be  distinctly  recognized. 

37.  To  add  one  primary  number  to  another  is  to  so  com- 
bine the  former  with  tlie  latter  that  in  the  resulting  num- 
ber each  unit  of  the  components  shall  retain  independence 
and  precisely  the  same  functional  relation  to  the  result 
(the  sum)  that  it  fulfilled  in  its  original  group.  The  con- 
cept is  so  immediate  to  that  of  primary  number  itself  (C/. 
the  specialization  of  various  manys,  %  2  et  seq.),  that,  while 
definition  is  appropriate  for  the  purposes  of  scientific  dis- 
course, it  hardly  admits  of  explanation.  The  numbers  are 
aggregated,  just  as  objects  now  thought  in  two  groups 
may  be  thought  in  one  group.  Also,  the  addition  of  any 
two  primary  niimbers  is  always  possible.  (But  note  that 
the  definition  is  only  for  primary  numbers,  vide  §  45.) 

38.  It  is  an  immediate  corollary  from   the    absolutely 

primary  theorem  of  number  {inde  §  13),  and  the  definition 

of  addition,  that  ,    ,        ,    , 

'  a  -^  h  =  0  -[-  a, 

that  is  to  say,  addition  is  a  commutative  operation.  The 
fact  is  called  the  Commutative  law  of  Addition.  It  obvi- 
ously extends  to  the  sum  of  any  number  of  numbers. 

39.  In  like  manner  the  addition  of  three  or  more  pri- 
mary numbers  is  associative,  that  is, 

(,,  _^  h)j^c  =  a^{h  +  r). 

This  fact  is  the  Associative  law  of  Addition. 

40.  An  algebraic  statement  like  the  foregoing,  the  truth 


PRIMARY   NUMBER. —  NUMERICAL   OPERATIONS.        45 

of  which  depends  on  the  very  nature  of  operations,  may- 
be called  a  formula,  as  distinguished  from  a  synthetic  equa- 
tion. In  a  formula  any  numerical  symbol  may  be  made 
particular  without  restricting  the  generality  of  any  other ; 
in  a  synthetic  equation  (e.g.,  a  -{-  h  =  c),  on  the  contrary, 
to  particularize  any  symbol  more  or  less  restricts  the 
meaning  of  every  other.  To  solve  a  synthetic  equation /or 
any  symbol,  means  to  find  a  definite  number  which,  sup- 
posing the  significance  of  every  other  symbol  known, 
substituted  for  the  unknown  symbol  will  satisfy  the  equa- 
tions ;  that  is  to  say,  make  of  it  a  formula  in  terms  of  the 
other  symbols.  The  name  identitij  is  often  used  for  for- 
mula as  here  defined.  When  it  is  necessary  to  distinguish 
between  a  formula  or  identity  and  a  synthetic  equation, 
the  sign  =  designates  the  former,  and  =  the  latter. 

41.  If  the  sum  of  tAvo  primary  numbers  and  one  of 
them  be  given,  the  other  may  be  formed  by  pairing  off 
every  unit  of  the  given  part  with  a  unit  of  the  sum,  and 
counting  the  unpaired  units  of  the  sum.  Since  addition 
is  commutative,  the  operation,  as  just  defined,  is  the  same, 
whichever  of  the  two  parts  of  a  sum  be  given.  Addition 
has  therefore  only  one  inverse,  called  Subtraction,  and  rep- 
resented by  the  minus  sign  (  — ). 

42.  The  problem  is  to  solve  for  x  the  synthetic  equa- 
tion —  ,  , 

a  -\-  X  =^  b. 

Counting  off  a  from  the  number  represented  by  each  mem- 
ber of  the  equation,  we  obtain  x  =  h  —  a\*  that  is  to  say, 

*  Of  course  the  common  notion  or  axiom,  '"if  equals  be  taken  from 
equals  the  remainders  are  equal"  is  here  involved.  But  truly  common 
notions  can  be  doubted  by  no  sane  man,  and  explicit  statement  of  uni- 
versal axioms  is  hardly  required  anywhere  except  in  systematic  treatises 
on  logic  or  epistemology. 


46 


NUMBER    AND    ITS    ALGEBRA. 


{h  —  a)  is  the  numbei-  wliicli  added  to  a  gives  h.  This 
number  {b  —  a)  is  called  the  remainder  or  diiference  result- 
ing from  the  subtraction  of  a  from  b.  Substituting  (b  —  a) 
for  X  gives  the  formula  — 

a  -\-  {b  ~  a)  =  b ; 

or,  by  the  commutative  law  of  addition, 

b  ~  a  -\-  a  =  b, 

which  is  the  formula  of  definition  of  subtraction. 

43.  Under  the  developed  concept  of  number,  any  chain 
of  additions  and  subtractions  enjoys  perfect  freedom  of 
commutation;  but  the  first  thing  to  strike  the  thoughtful 
student  in  subtraction  under  the  primary  concept,  is  the 
futility  of  seeking  general  laws,  because  the  operation  is 
possible  only  in  special  cases.  If  b  is  less  than  a,  b  —  a 
makes  no  sense. 

We  may  observe,  moreover,  that  provided  the  expres- 
sions mean  anything,  association  may  take  place  as  fol- 
lows :  a  —  m  —  n  =  a  —  (in  +  n)  ;  for,  adding  m  +  n  to 
each  member  of  the  equation,  we  obtain  a  =  a.  (Cf.  foot- 
note to  §  42.) 

In  like  manner,  a  -i-  b  ~  m  ~  u  =  a  —  (jii  -f  n  —  b)  = 
a  —  VI  —  (ji  —  b),  etc. 

Also  a  -\-b  ~  m  —  n  =  a  -{-  (I)  —  vi  —  ?;). 

This  is  the  ground  of  the  familiar  rule  about  "  signs  " 
and  parentheses.  Of  course,  the  rule  applies  only  under 
great  restrictions  to  primary  numbers. 

44.  In  practice  the  problem  often  occurs  to  find  the 
sum  of  a  number  of  equal  numbers  ;  e.g.,  how  many  shoes 
are  required  to  shoe  twelve  horses  ?  With  primary  num- 
ber this  is  only  a  special  case  of  addition.     It  was  a  true 


PRIMARY   NUMBER.  —  NUMERICAL   OPERATIONS.       47 

instinct,  however,  which  recognized  a  distinct  operation. 
But  the  instinct  was  too  often  disavowed  in  the  next 
breath  by  defining  multiplication  as  "  repeated  addition." 
Multiplication  with  primary  numbers  is  repeated  addition ; 
])ut  this  concept  is  incapable  of  development  without  doing 
great  violence  to  the  word  ''  repeated."  How  can  one  so 
define  multiplication,  and  then  say  V2xV3  =  V6? 
Xo  repeated  addition  can  attain  this  result.  This  is  antici- 
pating ;  but  nevertheless  we  may,  in  the  expectation  of  de- 
velopment, at  least  be  careful  not  to  prejudice  opinion.  If 
possible,  let  us  try  to  say  enough  to  define  multiplication 
for  primary  number  without  saying  so  much  that  the  way 
of  development  is  barred. 

45.  The  gradual  extension  of  the  meaning  of  terms  is 
perhaps  the  most  powerful  instrument  for  that  ordering 
and  simplification  of  knowledge,  that  transformation  of 
chaos  into  cosmos,  which  is  the  vocation  of  science.  The 
procedure  should  take  place  with  the  caution  befitting  its 
importance,  and  demands  at  every  stadium  a  consummate 
restraint  of  judgment  in  order  not  to  say  too  much.  The 
severest  self-criticism  alone  can  repress  the  tendency  of 
tyros  in  every  science  to  set  delimitations  which  confine 
development,  and  entomb  thought  in  empiricism. 

Note  carefully  that  even  addition  must  not  be  declared 
as  necessarily  increasing  a  number.  AVith  primary  num- 
bers a  number  is  increased  by  addition  ;  but  to  so  define 
would  bar  development.  jSTeither  in  general  does  multipli- 
cation increase  a  number,  nor  division  decrease  it,  and  to 
so  define  would  hide-bind  mathematics. 

46.  The  operation  of  multiplication  can  hardly  be  de- 
fined for  primary  number  without  prejudice  to  the  develop- 
ment so  necessary  to  mathematics,  pure  and  applied.     A 


48  NUMBER   AND   ITS   ALGEBRA. 

satisfactory  definition  has  never  been  framed ;  nor  must  it 
be  supposed  tliat  I  consider  the  feat  achieved  in  the  fol- 
lowing definition,  which  (or  something  like  it)  was  first 
offered,  I  believe,  by  De  Morgan.  The  matter  is  one  of 
paramount  importance  ;  for  all  rational  views  of  number 
have  been  developed  under  the  principle  of  the  persistence 
of  the  laws  of  the  operations,  addition,  multiplication,  and 
involution.  (Their  inverses  would  be  adequately  defined 
merely  as  such.)  Nor,  until  satisfactory  definitions  of 
these  operations  in  their  utter  generality  are  attained,  can 
the  fundamental  theory  of  the  subject  be  regarded  as  per- 
fect or  completely  established.  I  make  an  effort,  not  in 
contentment  with  the  result,  but  to  display  the  difficulties. 

The  Multiplication  of  any  number  by  another  consists 
in  affecting  the  former  (multiplicand)  in  precisely  the 
same  way  as  one  is  affected  in  the  other  (multiplier). 

Or,  in  Multiplication  one  number  is  so  combined  with 
another  that  one  of  them  shall  fulfil  in  the  result  the 
same  functional  relation  that  the  number  one  fulfils  in 
the  other. 

The  result  is  called  the  product. 

The  multiplication  of  any  t\yo  primary  numbers  is  al- 
ways possible. 

With  primary  numbers  the  foregoing  tentative  definition 
amounts  to  "  repeated  addition,"  nor  is  it  claimed  that  it  is 
much  better  as  a  scientific  achievement.  The  difference  is 
rather  pedagogical :  if  you  tell  a  pupil  that  "  multiplica- 
tion is  repeated  addition,"  he  is  disposed  to  think  he  fully 
understands  the  nature  of  the  operation ;  but  if  you  tell 
him  that  in  this  operation  the  multiplicand  is  affected  in 
the  same  way  as  one  is  affected  in  the  multiplier,  although 
he  will  not  at  first  receive  more  information  than  before. 


PBIMAIIY   NUMBER.  —  NUMERICAL   OPERATIONS.      49 

he  is  in  a  position  to  Avitlen  liis  concept  of  tlie  "  ways  "  in 
which.  07ie  may  be  affected  in  a  nnmber.  And  when  he 
recognizes  ratios  as  numbers,  and  that  any  number  is  its 
own  ratio  to  one,  the  composition  of  ratios  at  once  falls 
into  his  definition  of  multiplication.  (Vide  §§  80,  81.  82, 
83.)  In  other  words,  if  ultimate  development  is  not  prej- 
udiced by  the  definition  suggested,  it  is  only  on  the  score 
of  its  vagueness ;  since  in  each  new  extension  it  is  from 
the  principles  of  multiplication  itself  that  the  A\'ay  in 
which  07ie  is  affected,  or  its  functional  relation  to  the  mul- 
tiplier, can  be  comprehended.  Nevertheless,  it  may  be 
considered  that  (when  ratios  have  been  recognized  as  num- 
bers, and  therefore  necessarily  a  ratio  of  any  two  num- 
bers, and  any  number  as  its  own  ratio  to  o7ie'),  in  the  light 
of  the  independently  discovered  operation,  the  "  composi- 
tion of  ratios,'"'  *  the  definition  might  be  read  in  a  fuller 
sense  than  it  could  convey  to  a  beginner.  This  may  seem 
a  pitiful  plight  for  a  definition ;  but  I  can  only  point  out 
that  many  things  have  to  be  seen  to  be  understood,  that 
before  such  vision  they  must  in  their  very  nature  be  ''  unto 
the  Jews  a  stumbing-block,  and  unto  the  Greeks  foolish- 
ness." The  extended  meanings  of  multiplication  are  un- 
dreamed of  to  the  man  whose  only  notion  of  number  is  his 
abstraction  from  a  flock  of  sheep,  or  a  pile  of  coins. 

it  might  be  better  to  leave  the  numerical  operations  un- 
defined in  words,  and  in  the  case  of  multiplication  to  rest 
upon  its  commi;tative  and  associative  laws  (which  alone 
would  not  distinguish  it  from  addition),  and  its  law  of  dis- 
tribution with  addition,  as  at  once  governing  and  defining 
the  operation.     (For  these  laws,  vide  infra.) 

*  Cf.  Euclid,  Book  YI.,  23;  and  VI.,  def.  5;  or,  better,  IIalsted'.s 
Elements  of  Geometry,  §§  540,  544. 


50  NUIMBER    AND    ITS    ALGEBRA. 

It  may  be  objected  that  even  addition  has  been  defined 
only  for  primary  number ;  but  when  number  has  been 
seen  to  be  a  continuous  magnitude,  and  the  qualitative 
difference,  ^^osi'z^iye  and  negative,  revealed,  the  "common 
notion "  of  addition  immediately  applies.  And,  as  sug- 
gested in  Section  37,  to  define  addition  is  like  defining 
such  terms  as  more  or  less. 

The  only  point  at  all  recondite  is  that,  from  the  very 
clean-contrary  nature  of  positive  and  negative,  the  addi- 
tion of  a  negative  number  to  another  decreases  the  latter. 
Attention  is  called  candidly  to  this  generalization  of  addi- 
tion, as  well  as  to  the  application  of  greater  and  less  to 
negative  numbers  {Cf.  §§  116,  117,  198)  ;  but  the  propriety 
of  these  concepts  is  left  to  be  justified  of  their  fruits. 

47.  The  multiplication  of  primary  numbers  is  commuta- 
tive, i.e.,  — 

ah  =  ha. 

This  is  the  Commutative  Law  of  Multiplication. 

Its  truth  is  obvious,  for  three  rows  of  four  dots  in  each 
row  is  the  same  group  as  four  columns  of  three  dots  in 
each  column,  thus  — 


Also,  commutative  freedom  is  shown  to  extend  to  the 
factors  in  a  sefies  of  successive  multiplication,  i.e.,  — 

ahcde  =  ahedc,  etc. 

Multiplication  with  equal  generality  is  associative ;  that 
is,  any  group  of  factors  may  be  replaced  by  their  product, 
i.e,  ahcde  =  a(hcd)e. 

This  is  the  Associative  Law  of  Multiplication. 


PRIMARY   NUMBER.  —  NUMERICAL   OPERATIONS.      51 

48.  From  the  commutative  nature  of  multiplication,  it 
follows  that  when  a  problem  is  discerned  as  requiring  the 
multiplication  of  one  number  by  another,  it  never  makes 
the  slightest  difference  theoretically  which  is  taken  as  the 
multiplicand.  All  the  talk  about  carefully  distinguishing 
multiplier  and  multiplicand  so  prevalent  in  text-books  is 
sheer  nonsense.  If  you  wish  to  find  how  many  oranges 
you  must  provide  to  give  3  to  each  of  278  children,  it  is 
utterly  indifferent  whether  you  multiply  3  by  278,  or  278 
by  3.  As  a  matter  of  convenience  in  this  particular  ex- 
ample the  latter  process,  absurdly  decried  as  it  is,  is  the 
sensible  course.  The  problem  requires  the  combination  in 
multiplication  of  the  number  of  children  and  the  number 
of  oranges.  The  product  is  interpreted  as  a  number  of 
oranges.  In  neither  case  have  oranges  or  children  been 
multiplied  ;  processes  of  horticulture  or  procreation  would 
be  necessary  in  such  a  performance. 

Concrete  magnitudes  can  be  multiplied  by  numbers,  but 
such  processes  are  not  purely  arithmetical.  For  example, 
a  sect  of  a  straight  line  can  be  really  multiplied ;  but  the 
process  is  a  geometrical  construction.  Thus,  to  multiply  a 
sect  by  3,  lay  off  the  given  sect  three  times  in  a  straight, 
so  that  one  of  the  three  shall  lie  end-point  to  end-point 
with  the  other  two,  but  no  other  points  in  common.  The 
sect  between  the  non-coincident  end-points  is  the  required 
product.  It  would  be  an  easy  construction  to  multiply  a 
sect  by  ■y/2,  for  this  would  be  accomplished  by  the  fa- 
miliar process  of  '^  altering  "  it  in  the  ratio  of  the  diagonal 
of  any  square  to  its  side. 

Of  course,  it  is  perfectly  legitimate  to  speak  of  the  mul- 
tiplication of  concretes  in  the  sense  merely  of  an  interpre- 
tation  of   a  numerical  process.      Thus,   there  can  be   no 


■  62  NUMBER    AND   ITS    ALGEBRA. 

objection  to  saying  8  pounds  multiplied  by  152  make 
1,216  pounds  ;  but  it  is  utterly  mistaken  to  protest  against 
multiplying  152  by  8  in  performing  the  calculation. 

49.  Since  multiplication  is  commutative,  there  is  only 
one  inverse  problem  ;  viz.,  given  a  product  and  one  fac- 
tor, to  find  the  other.  The  operation  is  called  Division. 
Division  requires  the  solution  for  x  of  the  synthetic  equa- 
tion —  , 

ax  =  0. 

The  formula  of  definition  of  division  is  — 

a  {l>  I  a)  =  h. 

dotation  ally  a  line  laterally  *  presented  to  the  number 
symbols  (—  or  /  ),  a  colon  (  :  ),  or  a  combination  of  both 
(-f-)  represents  division.  The  first  is  generally  to  be 
preferred. 

50.  With  primary  number  division  amounts  to  repeated 
subtraction,  but  it  is  only  safely  defined  as  the  inverse  of 
multiplication.     (Vide  §  45.) 

51.  Under  the  developed  concept  of  number,  if  a  num- 
ber is  to  be  combined  with  a  series  of  others  which  operate 
successively  in  multiplication  and  division,  there  is  free 
commutation  and  association  in  using  the  operators  in  the 
manner  displayed  in  the  following  :  — 

(1)  (a  X  f^)  -^  c  =  (a  -i-  c)  X  b  =  a  X  h  /  c  =  a  -i-  c  /  b; 

(2)  {a  -~  h)  -i-  c  =  a  /  he  =  aj  c  -^  h  ; 

(3)  (ft  -V-  h)  X  {c  -^  d)  =  ac  j  bd  =  a  j  d  -~  b  /  e  =  ac  j  h 

-^  d,  etc. ; 

(4)  {fi  ^  b)  -^  (c  -^  d)  ^  ad  jbc  =  a  I  c-^b-d  =  ad  /  b 

-f-  c,  etc. ; 

*  The  "  minus  "  sign  is  presented  endwise  to  tlie  number  .symbols. 


PRIMARY   KUMBER.  —  NUMERICAL   OPERATIONS.       53 

as  may  easily  be  proved  from  the  laws  of  multiplication 
and  the  definition  of  division,  to  be  true  for  primary 
number,  if  the  opei'ations  have  any  meaning  at  all. 

But  as  in  the  case  of  subtraction  (§  43),  it  is  vain  to 
attempt  generalizations  with  division  under  the  primary 
concept  of  number,  for  division  is  possible  only  in  par- 
ticular cases. 

Thus,  considering  that  the  primary  numbers  represented 
are  such  that  the  statement  (ct  X  h)  -^  c  makes  sense, 
(a  -^  c)  X  l^  ii^ay,  or  may  not,  have  meaning ;  e.g.  (3  X  4)  -v- 
6  makes  sense,  for  there  is  a  primary  number  which  mul- 
tiplied by  6  gives  12 ;  but  (3  -i-  6)  X  4  is  meaningless  in 
terms  of  primary  number,  for  there  is  no  primary  number 
which  multiplied  by  6  gives  3.  Again  (15  X  4)  -;-  6  is 
intelligible,  but  not  (15-4-6)  X  4 ;  since  no  primary  num- 
ber multiplied  by  6  gives  15. 

52.  If  a  sum  of  two  primary  numbers  is  to  be  miiltiplied 
by  a  primary  number,  the  product  is  the  same  as  the  sum 
of  the  products  of  each  summand  by  the  multiplier,  i.e.,  — 

(rt  -{-  h)  c  =  ac  -[-  be. 

For  4  rows  of  5  in  a  row  is  the  same  group  as  the  sum 
of  two  groups  each  of  4  rows,  2  and  3  respectively  in  a 
ruw,  thus :  — 


The  principle  evidently  extends  to  the  sum  of  any  num- 
ber of  summands,  and  is  called  the  Distributive  Law  of 
Multiplication  and  Addition. 

53.    If  the  multiplier  be  a  sum,  of  course  redistribution 

'univ.'ehsi'i 
California 


54  NUMBER   AND   ITS   ALGEBRA. 

will  display  the  final  result  as  a  sum  of  simple  products, 
e.g., 

{a  +  h)  {c  -\-  d)  =  a{r-\-  d)  -\.h  {c -\- d)  =  ae -\-  ad  +  he  -\-  hd. 

54.  If  eacli  one  of  a  number  of  factors  be  a  result  of 
mixed  addition  and  subtraction,  the  Distributive  Law  ap- 
plies, but  with  primary  numbers  only,  under  the  miserable 
restrictions  inherent  in  inverse  operations. 

55.  Also  a  series  of  additions  and  subtractions  is  dis- 
tributable with  a  divisor.  It  is  sufficient  to  give  formal 
proof  in  one  instance  :  — 

To  prove  {rt  -\- V)  -^  c  =  (a  -4-  r-)  -f-  (Ij  -f-  c). 

Now,  {(a  -\-  b)  -^  c]  c  =  a  -\-  b  hy  definition  of  division. 

Again,  {(a  -^  c)  +  (i  -f-  <-)}  0  =  (a  ^  c)  c  -\-  (b  -i-  c)  c  by 
distribution  of  multiplication  and  addition ;  but  this  last 
also  =  a  -{-  b  hj  definition  of  division, 
.:  (a -\- b)  ^  c  =  (a -h  c) -\-  (b  ^  c).     (Cf.  foot-note  to  §  42.) 

56.  If  factors  be  sums,  redistribution  is  possible,  since 
the  original  case  merely  recurs  (vide  §  53)  ;  but  if  a  divisor 
be  a  sum,  it  cannot  be  distributed. 

(a  -\-b)^  (e-\-  d)  =  a  -  (^  +  d)  +  //  -  (.  +  d), 
but     a  -i-  (c  -)-  d)  does  not  equal  (a  -^  c)  -f-  (a  -f-  d), 
as  the  student  may  easily  satisfy  himself. 

Let  this  truth  emphasize  the  principle  that  all  such 
questions  are  matters  of  fact,  and  not  to  be  convention- 
ally decided. 

57.  It  frequently  occurs  in  practice  that  it  is  required  to 
repeatedly  multiply  a  number  by  itself.  Given  the  basal 
number  and  the  number  of  times  it  is  to  occur  as  a  factor, 
the  process  is  completely  determined.  The  original  num- 
ber is  called  the  base ;  the  number  of  times  it  is  to  occur 
as  a  factor  is  called,  according  to  the  point  of  view,  the 


PRIMARY  NUMBER.  —  NUMERICAL   OPERATIONS.       55 

exponent  of  the  base,  or  the  logarithm  of  the  result  to  the 
specific  base.  The  result  is  called  the  power.  The  expo- 
nent is  sometimes  called  an  index. 

Numbers  in  this  relation  are  notationally  represented 
thus  :  4^  =  64,  or  «*  =  e,  where  a  is  base ;  h,  the  expo- 
nent ;  c,  the  power.  The  phrase  logarithm  of  c  to  base 
a  is  written  in  algebraic  shorthand  thus,  log„  c. 

58.  When  the  exponent  is  two,  the  power  is  commonly 
called  the  "  square ;  "  and  when  the  exponent  is  three,  the 
"cube."  These  names  refer  to  true  and  proper  geometrical 
applications  of  number,  but  have  no  doubt  had  their  share 
in  postponing  general  recognition  of  number's  real  nature. 
{Cf.  §  25,  and  §§  230,  231.) 

59.  The  operation  of  combining  two  numbers  in  the 
sense  represented  notationally,  as  above  explained,  by  a*, 
is  called  Involution.  But  just  as  we  restrained  ourselves 
from  prematurely  regarding  multiplication  as  repeated  ad- 
dition, we  must  prejudice  no  subsequent  questions  by 
regarding  involution  as  repeated  multiplication.  It  is  re- 
peated multiplication  for  primary  numbers ;  but  when  we 
discern  other  modes  of  number  we  shall  see  that  such  is 
by  no  means  the  essential  nature  of  the  operation. 

60.  It  is  impossible  (for  me)  to  frame  a  definition  of 
involution  in  terms  of  primary  number  which  will  satis- 
factorily connote  the  simplest  and  the  general  meaning  of 
the  operation.  {Cf.  §§  45,  46.)  In  lieu  of  something  more 
satisfactory  I  make  the  following  attempt :  Involution  is  a 
combination  of  two  numbers  such  that  the  base  shall  appear 
factorially  in  the  result  in  a  mode  corresponding  to  that 
in  which  unity  exists  additively  in  the  exponent.  While 
this  definition  expresses  primary  involution,  it  is  not  in- 
consistent with  ultimate  meanings.     For  example,  if  unity 


56  NUMBER   AND   ITS   ALGEBRA. 

exists  three  times  aclditively  in  the  exponent,  the  base 
must  appear  three  times  factorially  in  the  power  ;  yet  when 
niimbers  are  conceived  in  which  unity  fulfils  a  relation 
the  inverse  of  primary  addition,  we  need  not  be  surprised 
to  discover  that  the  base  appears  in  the  power  in  a  relation 
the  inverse  of  that  of  a  direct  factor. 

[  «3  =  aaa,  and  a~^  =  -.-.-] . 
\  a    a    aj 

Again,  when  a  number  such  as  the  ratio  1  /  3  is  discerned, 
it  becomes  a  development,  not  a  recantation  of  former  opin- 
ion, to  discover  that  the  exponent  1/3  imposes  upon  a  base 
an  operation  which  shall  cause  it  to  appear  in  the  result  as 
one  of  three  equal  factors  of  itself,  since  1  /  3  is  one  of  three 
equal  summands  of  1.        (a*  =  V«-) 

It  would  be  anticipating  too  much  to  carry  testing  any 
further.  I  set  forth  the  definition  merely  as  the  best  that 
I  can  offer.  Perhaps  the  most  scientific  attitude  in  the 
dilemma  is  merely  to  note  the  sense  of  involution  for  pri- 
mary number,  alertly  waiting  to  discover  what  its  nature 
may  be  as  deepening  insight  reveals  other  modes  of  num- 
ber, and  surmising  upon  general  grounds  that  if  a*  means 
repeated  multiplication  when  i  is  a  primary  number,  it 
will  not  have  this  meaning  if  b  is  not  a  primary  number. 

61.  I  have  dwelt  upon  this  matter  because  it  is  an 
exceedingly  important  point.  The  application  here  of  the 
Principle  of  Continuity  {vide  §  103)  has  led  to  un- 
dreamed-of advances,  not  only  in  the  mathematics,  but 
in  the  physical  sciences. 

62.  Involution  is  evidently  not  commutative  :  «*  is  not 
!)"■.     A  unique  case  is  commutative  ;  2*  =  4^. 

Neither  are  successive  involutions  associative :  a/^^  is 
not  equal  to  ((I'^y. 


PRIMARY   NTTMBER.  —  NUMERICAL   OPERATIONS.      57 

63.  Let  the  student  find  the  difference  between  2(2==') 
and  (2-^)  (='). 

64.  "Law  of  Indices."  —  For  primary  numbers  it  fol- 
lows immediately  from  the  definition  (let  the  student  deduce 
the  forms,  however)  that  oVa"^  =  «*  +  ^  +  '';  (a^y  =  a^'^, 
and  a'^b''  =  (aby. 

Also  iib>  c,  rt*  ~r-  a'  =  a^''.     (See  also  §§  158,  191.) 

65.  Because  involution  is  not  commutative  there  are 
two  inverse  operations,  requiring  respectively  the  solution 
for  X  of  the  synthetic  equations  (1)  a'«  =  b,  and  (2)  a-^  =  //. 

66.  Operation  (1)  is  called  Evolutiox,  or  finding  the 
ath  root  of  b.  In  algebraic  shorthand  the  rtth  root  of  b  is 
written  -\^b.  The  radical  sign  is  derived  from  the  letter  r 
(radix).  In  actual  computation  (arithmetical  or  alge- 
braical) after  the  theory  of  exponents  has  been  general- 
ized, it  is  far  better  to  employ  indices  than  radical  signs. 

67.  Operation  (2)  is  called  Finding  the  Logarithm 
{aide  §  35). 

68.  The  Formula  of  Definition  of  Evolution  is  (V^)«  =  b. 

69.  The  Formula  of  Definition  of  finding  the  Logarithm 
is  a^^-a*  =  b. 

70.  As  has  been  seen  to  be  the  case  with  all  inverse 
operations  in  terms  of  primary  number,  these  inverses  of 
involution  are  evidently  possible  only  in  very  special  cases. 

71.  With  the  discovery  of  the  seven  operations,  and  their 
laws,  Commutative,  Associative,  Distributive,  and  the  Law 
of  Indices  or  Exponents,  the  foundation  of  arithmetic  and 
the  algebra  of  number  is  complete.  I  repeat  (Cf.  §§  20,  23) 
these  laws  could  never  have  originated  arbitrarily,  or  as 
springing  essentially  from  the  algebra.  As  ^'  algebraic 
laws  "  they  must  be  merely  the  expression  of  actual  prop- 
erties and  relations  of  number. 


68  NUMBER   AND   ITS    ALGEBRA. 

VIII.    Devices   of  Computation. 

72.  Various  devices  of  computation,  of  more  or  less 
practical  utility,  are  familiar  to  all ;  but  it  will  be  clear 
upon  any  thoughtful  consideration  that  they  possess  none 
of  the  fundamental  importance  suggested  by  the  promi- 
nent role  they  play  in  ordinary  text-books.  AVhat  is  usu- 
ally set  forth  as  a  general  exhibition  of  addition  must 
be  seen  to  be  several  partial  additions  and  a  convenient 
association  of  resulting  summands.  The  same  numbers 
would  have  their  parts  differently  associated  to  suit  dif- 
ferent notations,  e.g.,  XXXVII  -f  XXXVIII  =  LXXV ;  or 
37  +  38  =  75. 

The  average  high-school  graduate  labors  under  the  im- 
pression that  his  fashion  of  "  multiplying  "  is  essential  to 
the  matter,  and  arises  from  the  very  nature  of  things.  In 
"  division  "  he  learns  what  he  sometimes  regards  as  two 
ways,  ''Short'"'  and  ''Long."  The  names  are,  in  truth, 
appropriate  enough,  for  the  sole  difference  is  that  more  of 
the  necessary  thought  is  actually  written  down  in  the 
Long  than  in  the  Short  way.  Yet  the  abbreviated  form  is 
taught  first,  and  the  pupil  fancies  he  is  learning  some- 
thing new  and  more  difficult  when  he  learns  "  Long  divis- 
ion." 

The  rational  method  would  be  to  teach  first  an  expres- 
sion still  longer  than  the  "Long";  then,  as  skill  and 
power  of  retaining  conclusions  in  mind  increase,  conve- 
nient abbreviations  should  be  explained  and  recommended. 

73.  Let  the  student  critically  examine  his  habitual 
ways  of  "adding,"  "subtracting,"  "multiplying,"  and  "di- 
viding "  primary  nunil)ers,  bf)t]i  in  the  common  algorithm 
of  arithmetic,  and  algebraically.      Let   him   denote  every 


DEVICES   OF   COMPUTATION.  59 

act  of  his  mind  in  each  process  as  an  addition,  subtraction, 
multiplication,  division,  commutation,  association,  or  dis- 
tribution of  numbers,  under  the  definitions  and  laws  set 
forth  in  the  preceding  chapters.  To  take  a  very  simple 
example:  {an^  c^)  {a^  h^  c"^)  ~-  {aH>^  c^"-) 

=  (a^  a^  ¥■  V^  c^  c")  -f-  («•*  b^  c^^)   ...  by  association  and 
commutation. 

=  (a^h^c^^)  -=-  (a*b^c^^)  ...  by  three   partial   multi- 
plications by  law  of  indices. 

=  (o^/a*)  (P  /h^)  (c^^  I  c^°)  ...  by    association    and 
commutation. 

=  a^  ^^  c  .  .  .  by  three  divisions  by  law  of  indices. 
74.  Explain  how  a  multiplying  machine,  which  can 
do  no  more  at  one  time  than  multiply  a  number  of  ten 
places  by  another  of  ten  places,  may  be  used  to  multiply 
13693456783231  by  46381239245932. 
.  75.  The  involution  of  primary  numbers  may  be  ac- 
complished merely  by  repeated  multiplication.  As  soon, 
however,  as  one  investigates  logarithmic  series,  and  the 
construction  and  use  of  Tables  of  Logarithms,  he  learns 
command  of  a  more  facile  waj^  of  performing  this  labo- 
rious operation.  Before  learning  the  use  of  logarithms, 
one  ought  to  demand  good  wages  for  the  toil  it  would  cost 
him  to  find  9^" ;  afterwards  it  becomes  the  Avork  of  a  few 
minutes. 

76.  Evolution,  as  we  have  seen,  is  only  occasionally  pos- 
sible under  the  primary  concept  oi  number  ;  but  even  in 
the  simplest  of  these  possible  cases  the  device  of  calculation 
familiarly  used  by  the  high-school  pupil  is  rarely  under- 
stood, else  he  would  be  able  to  find  (however  laboriously) 
the  fifth  root  as  well  as  the  third.  Of  course  evolution  is 
too  laborious  to  be  carried  to  any  extent  until  Logarithmic 


60  NUMBER   AND   ITS   ALGEBRA. 

Tables  are  comprehended,  Avhen  it  becomes  easy.  But  if 
one  understood  how  his  device  for  extracting  a  second  or 
third  root  was  invented,  he  coukl  on  occasion  make  his  own 
rule  for  finding  a  fifth  root.  Let  us  investigate.  Properly 
distributing  and  associating,  it  is  seen  that  — 

{a  J^  hf  =  a-  -\-  b  {2  a -\-  h). 

Also  {a-{-h  -\-  c)-  =  (a  +  i)-  +  c  {2  (a  +  Z-)  +  c],  etc. 

Here  is  declared  a  rule  for  the  evolution  of  a  second  root 
of  a  number  ;  for  a  specific  composition  of  the  power  is 
displayed  in  a  way  to  make  decomposition  easy.  Likewise 
the  formulae  for  the  evolution  of  a  cube  root  are 

(a  +  by  =  a^  +  Z.  (3  a^  _^  3  ab  +  b''), 

and  (a-\-b  -^  cf  =  (ci  ^  b'f  J^  c  {^  {a -^  bf  +  3  (a  +  b)  c 
-f-  c-},  etc. 

In  exactly  the  same  way  the  formula  for  the  evolution  of 
a  fifth  root  is 

(ft  +  bf  =  «5  ^Jj(pa^j^l0  an>  +  10  a%''  +  o  ah^  +  b'),  etc. 

Suppose  the  fifth  root  of  33554432  is  required. 

liow  the  preceding  formulse  show  that,  if  the  root  be 
considered  as  the  sum  of  three  numbers,  the  corresponding 
power  of  the  sum  of  the  first  two  is  to  be  taken  away,  and 
the  remainder  decomposed  to  reveal  the  third  summand  of 
the  root,  and  so  on.  Therefore  we  could  not  go  wrong 
even  by  choosing  parts  of  the  root  at  random.  But  a  con- 
sideration of  the  arithmetical  notation  may  save  much 
trouble ;  for  it  is  plain  that  a  fifth  root  of  the  number 
before  us  has  two  digit  figures,  that  is,  it  is  to  be  regarded 
as  the  sum  of  a  number  of  tens  and  a  number  of  ones.  We 
compute  as  follows  :  — 


FIRST   EXTENSION    OF    THE   NUMBER-CONCEPT,      61 


a        b 

33554432 
24300000 
9254432' 

30  +  2 

a5_ 

5  «"     =  4050000 

(Here  we  guess  our  6 ;  the  calculation 
will  test  accuracy.) 

10  a^b   =    540000 

10  a%'-  =      36000 

5ab'   =        1200 

b^  =            36 

4627216 

9254432 

(Got  by  multiplying  4627216  by  &,as  the 
formula  directs.) 

77.  ISTow  let  the  student  compute  again,  taking  20  for  a 
and  12  for  b.  Also  let  him  prove  12  a  cube  root  of  1728, 
taking  6,  then  4,  then.  2,  as  summands  of  the  root. 


IX.    First  Extension  of  the  Number-Concept. 
Eatio.  —  Fractions.  —  Surds. 

78.  The  first  extension  of  the  concept  of  number  is  the 
identification  of  the  ratio  of  any  two  magnitudes  of  the 
same  kind,  and  without  qualitative  distinction  for  the  pur- 
poses of  the  comparison,  as  a  number. 

79.  This  step  was  taken  long  ago  (^Cf.  Introduction, 
p.  11),  and  is  now  universally  accepted  as  a  dictum,  even 
where  not  clearly  discerned  as  a  matter  of  insight. 

80.  This  development  of  the  number-concept  was  no 
doubt  occasioned  in  the  history  of  human  experience  by 
problems  of  practical  measurement.  {Cf.  Introduction, 
p.  13.) 

Thought  must  have  operated  as  follows :  If  the  numeri- 
cal relation  of  a  yard  to  a  foot  is  3,  surely  there  is  a  num- 
ber denoting  the  relation  of  a  yard  to  two  feet,  and  of  a 


62  NUMBER   AND   ITS   ALGEBRA. 

foot  to  a  yard.  That  is,  numbers  which  are  fractions  (vide 
§  83)  of  primary  number  were  discerned.  This  advance 
still  leaves  number  discrete,  that  is,  increasing  per  saltum. 
But  again,  as  a  second  step,  if  there  is  a  numerical  relation 
between  two  magnitudes,  one  of  which  is  a  fraction  of  the 
other,  surely  there  must  be  a  numerical  relation  between 
an 3^  two  magnitudes  of  the  same  kind,  even  though  neither 
be  a  fraction  (vide  §  83)  of  the  other.  Thus,  when  it  is 
proved  that  the  diagonal  and  side  of  a  square  are  absolute- 
ly incommensurable  {Euclid,  Book  X,  117),  the  mind  can- 
not tolerate  the  thought  that  a  numerical  relation  would 
exist,  provided  the  diagonal  were  just  the  least  bit  shorter, 
yet,  de  facto,  does  not  exist.  This  thought,  I  repeat,  is  in- 
tolerable. Moreover,  since  the  ratio  of  a  yard  to  a  foot 
is  an  exact  number,  surely  the  ratio  of  a  metre  to  a  foot  is 
exactly  whatever  it  is.  It  is,  of  course,  well  known  that 
the  metre  and  foot  are  incommensurable 

81.  The  connotation  of  all  ratio  (fractional  and  surd)  as 
number  evidently  makes  number  continuous  one  ivay,  to 
use  a  space  metaphor  on  account  of  the  exigencies  of  lan- 
guage. Thus,  under  this  concept,  number  begins  with  a 
ratio  smaller  than  any  assignable  fraction  of  1,  increases 
continuously,  passing  through  all  the  discrete  stages  of 
primary  number,  to  a  ratio  greater  than  any  assignable 
primary  number. 

82.  To  illustrate:  Start  with  the  ratio  of  the  weight 
of  these  pages  to  the  weight  of  a  granite  bowlder.  We 
begin  either  with  a  very  small  fraction  of  1,  or  a  surd 
smaller  than  a  very  small  fraction  of  1  (as  the  weights  are 
commensurable  or  not,  probability  being  vastly  in  favor  of 
the  latter  case).  Now,  by  gradual  abrasion  of  the  bowlder, 
decrease  its  mass;  the  ratios  of  the  weights  increase  con- 


RATIO.  —  FRACTIONS.  —  SURDS.  63 

tinuously  until  they  reach  1.  Continue  the  abrasion,  and 
the  ratios  increase  continuously,  passing  through  2,  3,  4, 
etc.  At  length  when  the  bowlder  has  been  reduced  to  a 
grain  of  sand,  the  ratio  will  be  greater  than  some  high 
primary  number. 

83.  The  foregoing  discourse  presumes  sufficient  familiar- 
ity with  the  subject  to  insure  the  reception  of  the  terms 
employed  in  their  precise  meaning ;  yet  it  may  be  service- 
able to  set  forth  the  following  definitions  {Cf.  §  205)  :  — 

(1)  Multiple.  —  One  magnitude  is  a  multiple  of  an- 
other when  the  former  may  be  separated  into  equal  parts, 
each  equal  to  the  latter.  (Of  course  ''  multiple  "  includes 
the  limiting  case  where  the  ''part"  is  the  whole,  i.e.,  jiiulti- 
plication  by  1.  It  is  merely  an  imperfection  of  language 
which  might  seem  to  exclude  this  case.) 

(2)  SuBMULTiPLE.  —  lu  (1)  the  "  latter  "  is  a  submulti- 
ple  of  the  ''  former." 

(3)  Fraction.  — Any  multiple  of  a  submultiple  is  a 
fraction.  (Of  course  if  a  is  a  fraction  of  J,  i  is  a  fraction 
of  a  ;  also  a  multiple  of  a  submultiple  may  reduce  either 
to  submultiple  or  multiple.) 

(4)  Commensurable.  —  Two  magnitudes  are  commen- 
surable if  either  is  a  fraction  of  the  other  ; 

(5)  Incommensurable.  —  if  neither  is  a  fraction  of  the 
other. 

(6)  Ratio.  —  That  definite  (exact)  numerical  relation 
{Cf.  §  80)  of  two  magnitudes  of  the  same  kind,  in  virtue 
of  which  one  is  either  a  fraction  of  the  other,  or  greater 
than  one  and  less  than  another  fraction  of  the  other, 
which  differ  as  little  as  we  please,  is  called  the  ratio  of 
the  former  to  the  latter. 

Of   course,    from    the   very  concept  of  ratios,   and    the 


64  NUMBER   AND   ITS   ALGEBRA. 

continuity  of  possible  ratios,  the  ratio  of  the  first  of  two 
magnitudes  to  the  second  is  greater  than  the  ratio  to  the 
second  of  any  magnitude  less  than  the  first.  Also  two 
ratios  are  equal  if  every  numerical  fraction  greater  than 
either  is  greater  than  the  other,  and  less  than  either  is 
less  than  the  other. 

A  ratio  is  often  spoken  of  as  "incommensurable,"  of 
course  as  an  abbreviated  expression,  since  it  takes  two 
things  to  be  incommensurable.  You  might  as  well  say,  "  x 
is  equal,"  as  to  say  "x  is  incommensurable."  The  abbre- 
viation is  for  incommensurahle  with  1.  Incommensurable 
ratios  may  be  called  surds. 

Let  it  be  clearly  noted  that  a  multiple,  a  submultiple, 
or  a  fraction  of  any  magnitude,  is  another  of  the  same 
kind :  but  that  the  ratio  of  two  is  a  number.  Thus  a 
fraction  of  a  time  is  a  time,  of  a  surface  a  surface,  of  a 
solid  a  solid.  Bvxt  the  ratio  of  one  solid  to  another  is 
a  number,  —  in  this  case  called  the  volume  of  the  former 
with  respect  to  the  latter. 

Note  also,  any  number  may  be  regarded  as  its  ratio  to 
1,  and  that  all  numerical  fractions  are  ratios,  but  not  all 
ratios  are  numerical  fractions. 

In  illustration  of  the  definition  of  a  ratio,  and  its  nota- 
tion, if  of  incommensurables,  consider  the  yard  and  the 
metre.  Measurement  {vide  §  203}  not  excessively  refined, 
gives  the  number  0.9143  +  for  the  ratio  of  a  yard  to  a 
metre.  This  is  to  be  understood  to  mean  that  a  yard  is 
greater  than  -fVVotT  ^^  ^  metre  and  less  than  tVoVo-  Meas- 
urement more  refined  would  yield  a  numerical  fraction  still 
more  closely  approximating  the  ratio.  The  ratio  in  ques- 
tion has  been  found  to  be  greater  than  0.914392,  and  less 
than  0.914393. 


OPERATIONS   UNDER   FIRST   EXTENSION.  65 

(7)  Surd.  —  Of  the  one-icay  continuous  Kumber,  the 
concept  of  "which  we  have  now  attained,  those  numbers 
which  are  incommensurable  with  1  may  be  called   surds. 

It  is  matter  of  discovery  that  the  V^  is  incommensur- 
able or  a  surd. 

The  term  surd  is  sometimes  exclusively  referred  to  the 
results  of  such  operations  as  V2 ;  but  Newton's  use  is 
a  philosophical  one.  For  the  V2  is  found  out  to  be 
1.41421  -f-,  that  is  a  number,  no  fraction  of  one,  but  greater 
than  1.41421,  and  less  than  1.41422,  which  is  surely  a 
number  of  precisely  the  same  kind  as  the  ratio  of  a  j^ard 
to  a  metre,  or  of  a  circle  to  its  diameter  (0.914392  -)-  and 
3.14159  -}-  respectively).  Incommensurable  numbers  re- 
sulting from  evolution  may  be  distinguished  as  radical- 
surds,  or  simply  radicals.     (Vide  §  145.) 

X.  Significance  of  Operations,  and  Special  Opera- 
tional Devices,  Appropriate  to  the  First 
Extension   of    the   jS^umber— Concept. 

84.  Euclid  probably  never  clearly  unified  his  concepts 
of  ratio  and  number;  but  following  Euclid  (q.v.,  and  cf. 
Halsted's  Elements  of  Geometry),  it  may  be  shown  that 
there  is  a  combination  of  ratios  which  obeys  the  same 
laws  that  govern  the  addition  of  primary  numbers,  or  of 
fractions  of  concrete  magnitudes,  an  inverse  operation 
corresponding  exactly  to  subtraction;  another  operation 
(''  composition  of  ratios  "),  which  obeys  the  same  laws  as 
the  multiplication  of  primary  numbers,  and  an  inverse 
("altering"  a  magnitude  in  a  given  ratio),  corresponding 
to  division. 

But,  from  the  very  definition  of  a  submultiple  of  any 


66  NUMBER    AND    ITS    ALGEBRA. 

magnitude,  the  finding  of  a  siibmultiple  is  identified  as  an 
operation  of  division,  since  the  problem  is  to  find  a  magni- 
tude which  imdtijylied  produces  the  given  magnitude.  Now, 
when  number  has  been  discerned  as  a  magnitude,  these 
reflections  make  it  plain  that  a  fraction  of  a  number  is  the 
number  resulting  from  the  division  of  that  number  liy 
another,  that  one-half  of  1  is  1  -f-  2,  etc.*  Also,  when  ratios 
have  been  identified  as  numbers,  and  number  thus  be- 
comes one-way  continuous,  the  operational  significance  of 
the  principles,  established  in  Chapter  YII.  for  Addition 
and  Multiplication  and  their  inverses,  extends  to  all  num- 
ber (primary,  fractional,  and  surd)  thus  far  conceived. 

Finally,  inasmuch  as  a  fractional  number  is  the  result 
of  dividing  one  primary  number  by  another,  it  may  be 
represented  most  conveniently  by  the  notation  already 
established  for  division.     (Vide  §  49.) 

It  would  be  impracticable  to  invent  individual  symbols, 
since  an  unending'  number  of  different  symbols  would  be 
demanded  to  designate  even  the  fractional  numbers  lying 
between  two  consecutive  primary  numbers  ;  nor  could  any 
such  symbol  be  used  otherwise  than  as  a  record,  since  in 
any  calculation  with  fractions  it  is  the  generating  numbers 
which  are  utilized,  and  not  the  fractional  number  itself. 

85.  It  seems  to  me  that  there  is  no  way  substantially 
different  from  the  lines  of  thought  I  have  followed,  where- 
by one  can  really  understand  what  he  is  doing  in  the  oper- 
ation 7/8  X  9/5  for  instance.  Teachers  of  arithmetic 
would  do  well  to  ponder  their  methods  at  this  point. 

*  The  only  explanation  ( ?)  of  such  conclusions  to  be  found  even  in  the 
splendid  Text  Book  of  Algebra  by  Professor  Chrystal,  is  "  the  statement 
that  /i  X  K  isK  oi  %  is  merely  a  matter  of  some  interpretation,  arithmeti- 
cal or  other,  that  is  given  to  a  symbolical  result  demonstrably  in  accord- 
ance with  the  laws  of  symbolical  operation."    Vol.  i.,  p.  VS. 


FRACTIONS.  67 

86.  It  remains  to  investigate  devices  for  performing  tlie 
seven  numerical  operations  in  this  extended  region  of  num- 
ber, and  in  t^vo  cases  to  discover  the  effect,  the  meaning,  of 
an  operational  combination;  viz.,  in  involution,  if  the  expo- 
nent be  a  fraction  or  a  surd.  In  the  first  place,  it  is  to  be 
borne  in  mind  that  it  is  one  thing  to  conceive  an  operation, 
and  another  to  perform  it.  For  example,  at  the  conclusion 
of  these  introductory  lectures,  it  will  be  plain  to  all  (if  now 
obscure)  thq,t  such  operations  as  involving  10  under  the 
exponent  tt,  or  finding  the  logarithm  of  5  to  the  base  12, 
are  perfectly  intelligible,  even  though  ignorance  of  loga- 
rithmic series,  or  of  the  use  of  a  table  of  logarithms,  should 
leave  one  without  devices  adequate  to  the  performance  of 
the  calculations. 

87.  It  should  be  observed  that  the  terms  iiumerator  and 
denoininator  applied  to  the  numbers  involved  in  a  numeri- 
cal fraction,  or  even  to  the  "terms"  of  a  ratio  of  incom- 
mensurables  (e.g.,  V2  /  6)  may  be  used  as  convenience 
suggests ;  but  conceived  operationally  they  are  to  be 
thought  as  dividend  and  divisor.  Tlie  numerical  symbols 
ill  tlie  algebra  of  this  chajjter  are  still  to  be  understood  as 
representinr/  2^^i'>^i(^fl/  ninnbers. 

88.  The  '•'  rules "  for  the  operations  of  addition,  multi- 
plication, and  division  of  fractions  follow  immediately  from 
the  definition  of  a  fractional  number,  which  is  merely  the 
recognition  that  the  inverse  of  multiplication  is  always 
possible,  that  the  result  of  the  division  of  any  primary 
number  by  any  other  is  a  number. 

Substraction  remains  refractory,  and  meaningless  unless 
the  minuend  be  greater  than  the  subtrahend. 

The  rules  are  only  the  generalization  of  Sections  51  and 
55,  q.v.,  yet  it  may  be  serviceable  to  discuss  them. 


68  NUMBEll    AND    ITS    ALGEBRA. 

89.    By  the  distributive  ]uw  — 

therefore  the  common  rule. 

Also  a  I  d  —  h  I  d  =  by  the   distributive   law,  if 

a  y  b  ;  therefore  the  common  rule. 

But  how  shall  we  perform  a  -{-  h  j  c,  or  a  j  h  -\-  c  j  d,  if 
a  /  h,  h  /  c,  and  c  /  d  are  fractions  ?  The  operation  is  dis- 
tinctly conceivable  ;  but  the  device  for  performing  it  re- 
quires an  intermediary  step  of  multiplication,  which  must 
therefore  be  investi.trated.     Consider  -^ 


'o*^ 


(1)  a  /  b  X  c  =  ac  /  b  =  a  -i-  h  /  c  =  c  -^  h  /  a. 

(2)  a/b  -^  c  =  a  I  bo  =  ac  H-  b. 

(3)  a/b  X  c/d  =  ac/bd,  etc.  (Cf  §51,  (3)). 

(4)  a/b  -i-  c/d  =  ad /be,  etc.  (Cf  §51,  (4)). 

(5)  a  /  b  =  a  /  b  X  c  /  c  =  ac  /  be,    also    a  /  b  =  {ci  J  h  -i-  c) 

ale 
X  c  =  -J—  . 
b  I  c 

(6)  a  X  b  I  c  =  ab  I  e  —  a  -^  c  I  b, 

all  by  the  laws  of  division  and  multiplication  (vide  §  51). 
Therefore  the  common  rules  :  From  (1),  To  multiply  a 
fraction,  multiply  the  numerator  or  divide  the  denomina- 
tor ;  from  (1),  to  multiply  by  a  fraction,  multiply  by  the 
numerator  and  divide  by  the  denominator  ;  from  (2),  to 
divide  a  fraction  multiply  the  denominator  or  divide  the 
numerator  ;  from  (1),  to  divide  by  a  fraction  divide  by  the 
numerator  and  multiply  by  the  denominator,  etc. ;  from  (3) 
and  (4)  for  cases  where  both  terms  of  the  operation  are 
fractions.  Also  from  (5)  it  is  obvious  that  to  multiply  or 
divide  both  terms  of  a  fraction  by  the  same  number  neither 
increases   nor  diminishes  it ;   and  from   (G),  the  result  is 


FRACTIONS.  69 

indifferent  whether  we  multiply  by  a  fraction,  or  divide  by 
its  reciprocal. 

90.  It  may  be  remarked  that  there  is  a  distinction  be- 
tween dividing  by  a  fraction  and  multiplying  by  its  recipro- 
cal, though  the  results  are  indifferent,  as  declared  in  Section 
89  (1).  The  operations  are  not  identical.  The  results  of 
4^,  4  X  16,  4  +  60  are  the  same,  but  the  operations  are  by 
no  means  identical,  b  /  a  is  called  the  recijjr'ocal  of  a  /  b, 
and  may  be  obtained  operationally  from  the  latter  by 
dividing  1  hy  a  /  b  ;  for  1  -i-  a  /  b  =  b  /  a.  Moreover  "  in- 
vert "  is  a  short-cut  term  which  may  be  used  among  those 
whose  knowledge  of  first  principles  is  assured ;  but  it 
should  never  be  used  in  explanation,  as  designating  an 
operation  —  one  can  as  little  turn  a  number  upside-down 
as  inside-out. 

In  the  United  States  of  America  the  custom  is  almost 
universal,  never  to  divide  by  a  fraction,  but  to  choose 
instead  the  equivalent  operation  of  multiplying  by  its 
reciprocal.  In  Europe  this  is  not  so  commonly  felt  to  be 
more  convenient.  As  a  question  of  practical  calculation 
the  matter  is  of  no  importance  ;  but  it  is  surely  lamentable 
if  pupils  are  led  to  think  that  they  are  dividing  by  a 
number  when  they  are  actually  multiplying  by  a  different 
number  of  such  relative  value  that  the  results  are  equiv- 
alent. Notationally  a  fractign  expressly  represents  an 
unperformed  operation.  The  unexpressed  result  is  the 
definite  number  :  thus,  7/6  means  7  divided  by  6 ;  and  the 
result  is  a  number  greater  than  1  and  less  than  2,  a  defi- 
nite value  of  the  continuous  magnitude  we  call  Number. 

A  fraction  in  operation  is  to  be  employed  as  a  composite 
term  consisting  of  a  dividend  and  a  divisor.  Now,  it  can 
be  reasonably  explained  even  to  a  very  young  student  of 


70  NUMBER    AND   ITS   ALGEBRA. 

aritlimetic  that  to  divide  by  a  quotient  is  equivalent  to 
dividing  by  the  dividend  and  multiplying  by  the  divisor. 
This  having  been  established,  he  can  see  that  the  problem 
to  divide  by  ajh  resolves  itself  into  dividing  by  a  and 
multiplying  by  b.  If  the  dividend  is  an  integer,  he  has 
simply  to  do  this.  If  the  dividend  is  a  fraction,  he  must 
first  have  been  led  to  see  that  a  fraction  is  rmiltiplied  by 
multiplying  its  numerator,  or  dividing  its  denominator ; 
and  divided  by  dividing  its  numerator,  or  multiplying  its 
denominator. 

If  these  principles  are  discerned,  he  can  proceed  in  any 
manner  he  prefers.  It  is  of  no  theoretical  consequence 
how  he  sets  down  on  paper  mental  conclusions.  There 
is  no  obstacle  to  performing  the  division,  under  the  princi- 
ples stated,  just  as  the  symbols  stand :  7/6-^3/5  = 
35/18. 

A  very  low  order  of  convenience  is  subserved  by  mak- 
ing a  different  problem  of  identical  result  :7/6  X  5/3  = 
35/18. 

This  discussion  may  seem  almost  trifling  ;  but  if  one  will 
reflect  that  the  average  common-school  pupil  thinks  he  must 
transform  any  such  problem  of  division  into  a  problem 
of  multiplication,  some  deficiency  in  the  usual  instruction 
at  this  point  will  be  apparent.  I  am  convinced  that  our 
schools  require  systematic, instruction  in  arithmetic  of  chil- 
dren entirely  too  young  to  be  capable  of  the  reasoning  and 
insight  demanded. 

In  such  cases  the  best  one  can  do  is  never  to  leave  any- 
thing totally  unreasonable  to  the  child.  Even  to  a  young 
child  very  recondite  matters  can  be  a  little  explained  — • 
brought  within  a  dim  light  of  reason,  if  not  clearly  illu- 
minated.     One  thing  is  certain,  —  bad  history,  bad  gram- 


FRACTIONS.  71 

mar,  bad   chemistry,  or  bad  mathematics,  is  always  bad 
pedagogy  as  well. 

If  instruction  in  so-called  arithmetic  is  always  to  have 
reference  to  concrete  magnitudes,  as  recommended  *  by  the 
latest  "psychology  of  number"  (Cf.  Introduction),  the 
simplest  method  for  division  by  a  fraction  would  be  to 
reduce  to  common  denominator :  thus,  7/6-i-3/5  =  35/ 
30  -^  18  /  30  =  35  / 18.  Indeed,  it  may  well  be,  when 
arithmetic  has  to  be  taught  to  children  too  young  for  the 
subject,  that  this  method  is  the  best  as  a  first  pi'ese7itation 
of  the  matter.  Because  the  crudest  notion  of  numerical 
fractions,  and  blindness  to  the  true  significance  of  our  nota- 
tion of  fractions,  is  not  incompatible  with  some  rational 
comprehension  of  this  process. 

91.  We  may  now  return  to  ovir  problems,  a  -\- h  [  c  and 
a  I  h  -{-  c  I  d.  By  Section  89  (5)  they  may  be  brought  under 
the  case  oi  a  I  d  -\-  b  j  d.      For  a  -{-  h  j  c  =  ac  /  c  -|-  b  /  c  = 

'^^_±A.      And  a/b-i-  c/d  =  ad/bd-\-  cb  /  db  ^^I^Al^. 

There  is  often  a  better  way  of  solving  the  second  prob- 
lem. Evidently  if  b  and  d  have  a  common  multiple,  vi, 
less  than  their  product,  it  would  be  advantageous,  es- 
pecially if  several  fractions  were  to  be  added,  to  reduce 
to  a  common  denominator  by  multiplying  both  terms  of 
each  fraction  by  m-divided-by-the-denominator.  No  doubt 
all  are  familiar  with  a  device  for  finding  the  least  common 
multiple  of  two  or  more  numbers.      (Vide  §  242.) 

92.  Inasmuch  as  an  exponent  of  involution  when  a 
primary  number  requires  merely  repeated  multiplication, 
we  see  — 

*  Psychology  of  Number,  McLellan  .and  Dewey,  p.  IIG. 


72  NUMBER    AND   ITS   ALGEBRA. 

(a  Jh)P  =  a  /h-a  /h-a  /h   •   •   •    =  av  J bP. 

Also,  since  V^  /  V^  •  Va  /  V^  r=  a  J  b, 

therefore,  V*  /  ^  =  V^/  /  V^,  etc. 

93.  If  a  fraction  is  expressed  as  a  sum  of  decimal  frac- 
tions, e.g.,  41.2164,  evolution  is  apparently  performed  pre- 
cisely as  in  Section  76.     This  is  permissible,  because — • 

41.2164  =  -VoVoV-  and  V-VoVoV"  =  V412164  --  VlOOOO. 
Our  notation  renders  it  easy  to  perform  a  portion  of  this 
calculation  at  a  glance  by  "  pointing  off;  "  but  the  operation 
must  be  understood  as  finding  the  •\/412164,  and  then  di- 
viding it  by  VlOOOO. 

Let  the  student  perform  the  calculation,  not  losing  sight 
of  ivhat  he  is  doing  in  how  he  does  it. 

Let  him  also  fully  express  the  operations  involved  in  the 
conclusion,  41.2164  =  -VoVoV-  ^^^^  notation  is  so  perfect 
that  it  may  almost  be  said  to  work  automatically,  and  for 
this  very  reason  it  often  blindfolds  teacher  and  pupil. 

It  would  richly  repay  the  student  to  perform  just  once  in 
his  life  such  a  calculation  as  V41.2164  under  an  imperfect 
notation.  Let  him  do  this,  expressing  everything  in  the 
Roman  characters. 

94.  As  has  been  said  (§  83  (7)),  it  is  a  matter  of  dis- 
covery whether  or  not,  in  any  particular  case,  Va  is  a  surd. 
(Cy.  §  156.)  For  example,  if  in  the  process  displayed  in 
Section  76,  it  appears  that  no  primary  number  is  the  root 
in  question,  we  may  go  on  in  the  process  of  Section  93,  and 
find  a  fraction  approximating  as  near  as  we  please  the  surd 
number  which  is  the  true  root.  Under  such  conditions  the 
root  is  a  surd,  and  the  process  described  interminable ;  but 
it  would  carry  us  too  far  afield  to  investigate  just  now 
general  criteria  for  deciding  whether  the  result  of  given 


FINAL   EXTENSION   OF   NUMBER-CONCEPT.  73 

combinations  of  given  numbers  is  a  commensurable  or 
incommensurable  number.  (  Vide  §  249.)  Let  the  student 
critically  examine  his  familiar  process  in  "  finding  V2." 

95.  In  general  an  incommensurable  number  cannot  ope- 
rate, or  be  operated  upon,  in  ultimate  calculation  in  com- 
binations with  primary  or  fractional  numbers.  In  lieu  of 
using  the  surd  itself,  we  must  use  a  fraction  differing  from 
it  by  as  little  as  we  please ;  e.g.,  if  the  ratio  of  a  circle  to 
its  diameter  enter  into  the  calculation,  we  employ  some 
approximate  fraction,  such  as  3.14159.  Su.rds  which  are 
roots  of  primary  numbers  or  of  fractions  may  operate  with 
their  exact  force  in  special  cases,  and  in  a  partial  way ;  e.g., 
(a/2)3  =  2;     V2  V3  =  V6  ;     2  Vl2  =  4  V3  ;     V2/3  = 

1/3  V6,  etc. ; 
but  investigations  into  such   combinations   must  be  post- 
poned to  the  next  chapter,  as  well  as  the  interpretation 
of  a%  if  s  is  a  fraction  or  a  surd. 

96.  Finally,  let  it  be  distinctly  recognized  that  the  great 
stumbling-block  which  confronts  us  at  every  turn  is  the 
wretched  limitation  to  special  cases  of  the  operation  the 
inverse  of  addition,  that  a  —  b  is  meaningless  if  a  <  b. 

XI.    Final  Extension  of  the  Numbek-Concept. 
Principle  of  Continuity. 

97.  Primary  number  is  a  discrete  magnitude.  The  first 
extension  of  the  number-concept  (the  connotation  of  ratios 
as  number)  made  number  one  way  continuous.  (^Vide 
§81.) 

The  conception  of  number  as  continuous  in  a  far  more 
general  sense  grew  from  the  application  of  a  principle,  at 
first  presented  as  an  assumption,  but  which  is  so  inces- 


It 


4  KUMBER    AND   ITS   ALGEBRA. 

santly  and  overwhelmingly  corroborated  that  its  rank  as  a 
genuine  and  compulsory  theory  is  perhaps  as  firmly  estab- 
lished as  that  of  any  scientific  jDrinciple  whatsoever. 

98.  As  has  been  repeatedly  shown  in  the  foregoing 
chapters,  the  combination  of  numbers  in  the  inverse  opera- 
tions is  meaningless  under  the  primary  concept  except  in 
special  cases.  For  example,  5  —  5,  5  —  6,  5  -f-  C,  V5, 
logs  6,  etc.,  result  in  no  primary  numbers  at  all. 

The  "  first  extension  "  gives  meaning  to  the  last  three  of 
the  cases  just  cited ;  for,  although  in  the  treatment  here 
presented,  a^,  where  s  is  fractional  or  surd,  was  not  inter- 
preted from  the  recognition  of  all  ratio  as  number,  the  true 
meaning  might  have  been  developed  at  that  point,  and 
logs  6  thereby  rendered  intelligible.*  All  this,  be  it  noted, 
without  understanding  5  —  5,  or  5  —  G  as  a  number,  or 
even  imagining  the  development  yet  to  come  after  this 
insight  is  attained. 

99.  For  centuries  science  rested  here,  either  not  regard- 
ing such  combinations  as  intelligible,  and  their  results  as 
numbers  ;  or  only  in  a  halting  fashion,  regarding  the  com- 
binations as  symbolic  jugglery,  and  the  results  as  ''  imagi- 
nary numbers."  And  at  the  present  day  it  is  only  by  the 
enlightened  van  among  men  of  science  that  this  stage  has 
been  passed. 

Negative  numbers  were  in  this  way  long  called  "  imagi- 
nary ; "  but,  as  they  gradually  forced  themselves  into  reluc- 
tant minds,  the  appellation  was  narrowed  to  denote  V— 1. 

100.  It  was  only  after  a  long  struggle  that  negative 
numbers  gained  recognition.     I  have  not  the  erudition  to 

*  It  was  deemed  more  convenient  to  take  the  final  step  at  once;  since 
the  principle  which  displays  ratio  as  nuniher,  and  the  general  iirinciple 
to  which  the  whole  treatment  converges,  are  really  one  and  the  same. 


PINAL   EXTENSION   OF   NUMBER-CONCEPT.  75 

furnish  exact  dates,  but  I  know  that  Cardan  in  1545  in 
his  Ars  Magna  calls  them  ^' numeri  Jicti;"  and  it  is  com- 
monly asserted  that  Descartes  in  the  seventeenth  century 
was  the  first  to  rend  this  portion  of  the  veil :  and  I  sup- 
pose that  those  who  half-heartedly  follow  in  the  wake  of 
science  continued  long  afterward  to  regard  negative  num- 
bers as  "  imaginary,"  and  all  operation  therewith  as  some- 
how a  trick  of  algebraic  signs  empty  of  numerical  meaning. 
Certain  it  is  that  such  is  the  attitude  even  to-day,  not,  it  is 
true,  of  those  who  follow  in  the  wake,  but  of  those  who  do 
not  follow  science  at  all,  though  engaging  a  large  share  of 
public  attention  as  teachers  thereof.  For  certain  also  it  is, 
that  at  the  close  of  the  seventeenth  century,  Newton  with- 
drew negative  numbers  (and  therefore,  as  will  duly  appear, 
zero,  and  positive  and  negative  infinity)  from  the  befogged 
region  of  "  nuvierl  ficti,"  and  revealed  them  as  "  numeri 
veri. 

The  last  stage  of  this  gradual  process  of  enlightenment, 
in  which  V  — 1  is  still  regarded  as  "  imaginary,"  is  yet  the 
stronghold  of  ignorance  of  fact,  of  prejudice,  and  of  color- 
blindness to  philosophic  evidence. 

101.  I  would  have  no  war  of  words  over  the  appellation 
''  imaginary."  The  term  in  this  connection  historically 
has  meant,  and  yet  baldly  means,  "  impossible,"  or  incom- 
prehensil/le.  Of  course  it  has  no  such  meaning  among  the 
best  mathematicians  of  to-day;  but  that  it  is  so  received 
by  the  unscientific,  by  many  teachers  of  mathematics,  and 
by  the  vast  majority  of  undergraduate  students,  cannot  be 
disputed. 

The  matter  of  a  change  in  terminology  is  not  of  prime 
importance,  for  terms  may  be  disassociated  in  technical 
use  from  their  general  meaning.     It  is  a  question  of  ex- 


76  NUMBER    AND    ITS    ALGEBRA. 

pediency.  While  sympathizing  with  the  conservative  who 
object  to  all  innovations  as  tending  to  confuse  the  vast 
literature  of  the  science,  neomo7i{c  is  so  much  more  appro- 
priate, and  '< imaginary"  or  "impossible"  so  misleading, 
that  the  benefits  of  the  change  appear  to  outweigh  the  in- 
conveniences. A  reformation  in  terminology  is  not  nearly 
so  confusing  as  changes  in  notation,  such  as  have  often  been 
brought  about;  for  example,  the  famous  propaganda  of 
"d-isni  versus  dot-age"  (dy  / dx  versus  y),  which  Dr.  Pea- 
cock began  while  yet  an  undergraduate,  in  league  with 
Herschel,  Babbage,  and  Maule.  The  reform  was  finally 
adopted  at  Cambridge,  and  Newton's  notation  soon  became 
entirely  excluded.  Nowadays  mathematicians  find  no  con- 
fusion in  using  both  notations. 

102.  The  principle  which  has  so  fruitfully  widened  the 
concept  of  number,  yielding  perfect  self-consistency  of 
number,  and  ever  deepening  adaptation  to  Nature,  I  call 
the  Principle  of  Continuity,  in  emphasis  of  its  most  im- 
portant outgrowth,  the  unlimited,  twofold  continuity  of 
number. 

This  principle  may  be  stated  as  follows  :  — 

103.  Principle  of  Continuity.  —  The  coynhination  of 
two  numbers  in  any  defined  operation  is  always  possible,  the 
result  real,  and  a  number  ;  and  the  precise  effica-cy  in  any 
operation  of  a  number  thus  revealed  is  determined  by,  and 
may  be  discovered  from,  the  formula  and  laws  of  definition 
of  til  e  operation  in  question. 

104.  Before  considering  details,  a  glance  at  the  results 
which  have  more  than  justified  the  postulation  of  this 
principle  may  be  useful  in  giving  the  student  the  proper 
perspective  of  the  subject. 

The  principle  at  once  makes  negative  numbers,  zero,  in- 


PRINCIPLE    OF    CONTINUITY.  77 

finities,  fractions,  surds,  neonionic  and  complex  numbers, 
all  equally  numbers.  Also  Number  thus  becomes  unlim- 
itedly  continuous  in  a  double  sense,  whereby  undreamed 
of  adaptability  to  Nature  is  revealed,  and  all  numerical 
operations  proceed  untrammelled  by  particularity. 

One  who  will  logically  apply  the  Fr'inciple  of  Continuity 
will  arrive  at  all  classes  of  numbers  —  or  divisions  of 
Number  —  with  equal  necessity  and  facility.  Negative  or 
fractional  numbers  will  appear  as  much  derived,  as  little 
original,  or  primary,  as  those  numbers  still  commonly 
called  "  irrational  "  or  •'  imaginary."  One  of  these  classes 
is  as  foreign  as  any  other  to  the  primary  concept  of  num- 
ber ;  that  is,  the  concept  of  number  as  discrete,  the  concept 
which  knows  only  one  number  between,  say,  5  and  7. 

If  the  symbol  i  be  set  apart  to  represent  the  neomon 
( V  —  1),  ■we  seem  to  have  in  the  expression  x  -f-  yi  the 
most  general  numerical  form  to  which  the  laws  of  number 
lead.*  For  it  has  appeared  upon  investigation  that  no 
combination  of  numbers  in  any  conceived  operation  can 
result  in  a  form  essentially  different.  Neither  has  any 
operation  essentially  different  from  the  seven  fundamental 
operations  developed  from  them.  It  might  be  surmised 
that  investigation  would -reveal  some  fourth  direct  opera- 
tion growing  out  of  involution,  as  involution  grew  out  of 
multiplication,  and  multiplication  out  of  addition ;  but  such 
does  not  seem  to  be  the  case.  No  ground  of  distinction  is 
furnished  for  a  new  species  of  operation.  That  is  to  say, 
the  operation,  if  assumed  ta  be  distinct,  would  show  itself 
not  essentially  so,  by  failing  to  lead  to  new  modes  of 
Number.     In  other  words,  if   the   investigations   referred 

*  For  this  expression,  a  complex  number  in  algebraic  form,  is  numer- 
ically neomouic  if  x  =  0,  aud  numerically  whatever  x  is,  if  y  =  0. 


78  NUMBER   A^D   ITS   ALGEBRA. 

to  are  trustworthy  (as  is  no  doubt  the  case),  there  can  arise 
no  new  opportunity  to  apply  the  Principle  of  Continuity, 
so  as  to  still  further  widen  the  meaning  of  ISlumber. 
Number  in  its  ultimate  sense  is  therefore  seen  to  form 
(what  primary  numbers  do  not,  nor  any  curtailed  concept) 
a  universe  complete  in  itself,  such  that  starting  in  it  we 
are  never  led  out  of  it.  Cayley  says,  whether  with  sound 
philosophy  or  essential  contradiction  of  terms  I  will  not 
attempt  to  discuss,  "  There  may  very  well  be,  and  perhaps 
are,  numbers  in  a  more  general  sense  of  the  term  (quater- 
nions are  not  a  case  in  point,  as  the  ordinary  laws  of  com- 
bination are  not  adhered  to)  ;  but,  in  order  to  have  to  do 
with  such  numbers  (if  any),  Ave  must  start  with  them.''  * 

105.  I  believe  that  very  few,  even  among  students  of 
mathematics,  are  aware  of  the  chaos  of  their  conception 
of  number,  in  spite  of  long  and  familiar  use. 

The  difficulty  here,  as  everywhere,  is  the  attainment  of 
true  concepts,  insight  into  the  principles  involved. 

I  balieve  that  the  present  condition  is  due  to  the  fact 
that  successive  generations  of  students  have  not  had  the 
difficulties  honestly  presented  to  them,  and  have  seldom 
even  considered  fundamental  theory.  They  have  been 
entrapped  into  .an  unwarranted  complacency;  they  have 
juggled  with  symbols  which  are  meaningless  to  them,  and 
for  the  most  part  without  even  noticing  that  no  concept 

*  From  note  made  long  ago ;  exact  reference  lost.  In  regard  to  qua- 
ternions it  may  be  observed,  tliat  though  in  their  ordinary  presentation 
certainly  not  numbers,  it  is  possible  that  they  may  yet  be  divested  of 
extra-numerical  properties.  Speaking  of  the  anomaly  according  to  which 
quaternions  in  the  common  interpretation  would  make  'A  niv-  negative, 
whereas  >^  ?;i  is  positive  and  tlie  wliole  positive,  Dr.  INIacfarlane,  in  his 
Algebra  of  Physics,  remarks,  "  If  this  is  a  matter  of  convention  merely, 
then  the  convention  in  quaternions  ought  to  conform  to  the  established 
convention  of  analysis;  if  it  is  a  matter  of  truth,  which  is  true  ?  " 


PRINCIPLE   OF   CONTINUITY.  79 

rises  "witli  the  words  tliey  utter,  tlie  symbols  they  write,  — 
tliat  their  discourse  upon  number  is  vox  et  i:)raeterea  nihil. 
I  believe  that  an  opposite  result  would  be  prevalent,  had 
an  opposite  course  been  pursued  by  teachers  and  authors, 
and  tliat  we  would  now  be  reaping  harvests  instead  of 
sowing  seed. 

106.  Some  one  ignorant  of  trigonometry,  of  the  ana- 
lytical treatment  of  geometry,  of  the  Calculus,  of  the 
varied  fields  of  applied  mathematics,  and  to  whom  the 
boundless  realms  of  pure  mathematics  loom  misty  and 
fantastic  —  some  such  one,  I  say,  may  ask,  "Why  all  this 
striving  to  make  number  continuous,  this  travail  to  pro- 
duce concepts  of  number  and  numerical  operations,  which 
shall  be  perfectly  general  and  unrestricted  ?  The  answer 
is,  the  need,  intellectual  and  practical,  is  urgent,  impera- 
tive. Establish  the  Principle  of  Continuity,  and  Arith- 
metic becomes  a  logically  perfect  universe,  and  besides, 
all  Xature  becomes  harmoniously  numerical ;  number  and 
its  laws  pervading  it  as  an  essential  principle.  Emerson's 
noble  lines,  in  which,  with  the  poet's  seer  gift,  he  speaks 
truer  than  he  knew,  then  become  literal  fact :  — 

"For  Nature  beats  in  perfect  tune, 
And  rounds  with  rliynie  her  every  rune  ; 
Whetlier  slie  work  in  land  or  sea, 
Or  hide  imderground  her  alchemy. 
Thou  canst  not  wave  thy  staff  in  air, 
Or  dip  tliy  paddle  in  the  lake, 
But  it  carves  the  bow  of  beauty  there. 
And  the  ripples  in  rhymes  the  oar  forsake  .  .  . 
Not  unrelated,  unaffied, 
But  to  each  thought  and  thing  allied 
Is  perfect  Nature's  every  part. 
Rooted  in  the  mighty  heart." 


80  NUMBER   AND   ITS   ALGEBRA. 

Besides,  the  assumption  has  been  made,  and  its  first 
fruits  are  the  attainments  of  the  physical  sciences  during 
the  last  two  centuries.  The  progress  in  exact  physical 
science  and  the  dependent  arts  has  been  due  to  the  power 
and  freedom  conferred  upon  analysis  by  this  postulate ; 
for,  as  I  have  said,  it  is  implicit  in  all  modern  analysis, 
even  whcTi  denied  with  the  mouth  of  the  calculator.  (See 
also  §§  110,  117.) 

107.  Like  all  profound  principles,  this  one  of  the  con- 
tinuity and  qualitative  distinctions  of  number  is  a  onatter 
of  insight,  and  does  not  admit  of  easy  demonstration.  One 
man  cannot  think  for  another  any  more  than  he  can  eat  for 
him ;  but  if  a  student  will  fix  alert  and  intelligent  atten- 
tion upon  the  inherent  development  of  the  idea,  and  upon 
the  manifold  Avitness  borne  by  almost  every  phenomenon, 
he  will  at  last  behold  the  Principle,  manifest  in  ten  thou- 
sand undreamed-of  relations. 

108.  Tt  is  not  practicable  to  give  more  than  one  example 
of  the  mental  attitude  I  desire  to  excite.  I  choose  one 
which  affords  a  double  illustration  :  in  the  first  place,  yield- 
ing a  geometrical  instance  of  the  Avay  in  which  concepts  in 
every  science  are  extended  to  conform  to  deepening  insight, 
an  extension  analogous  to  the  development  of  the  primary 
number-concept ;  and  in  the  second  place,  displaying  (as  a 
consequence  of  this  attainment  of  an  adequate  geometric 
definition)  an  impressive  discovery  of  supreme  law  —  pro- 
vided ratios  are  numbers,  and  number  positive  and  nega- 
tive —  in  what  seems,  to  nai've  observation,  utter  fortuity. 

109.  Illustration.  —  (1)  The  primary  concept  of  the 
division  of  a  sect  by  a  point  is,  of  course,  that  the  point  is 
on  the  sect ;  but  investigation  shows  that  a  widening  of  the 
concept  is  required  to  fit  facts  presented  by  Nature.     It  is 


PRINCIPLE   OF   CONTINUITY.  81 

discovered  that  if  any  point,  P,  in  the  straight  of  a  sect, 
AB  (on  or  out  of  tlie  sect),  shall  divide  it  into  the  segments 
FA  and  PB,  then  innumerable  theorems  only  partially 
true,  and  therefore  none  of  their  inverses  true  (^vide  infra), 
under  the  primary  concept,  become  universally  true,  and 
therefore  their  inverse  propositions  true,  under  the  extended 
concept. 

(2)  The  same  term,  division,  is  necessarily  retained  for 
this  new  relation ;  for  it  is  the  very  essence  of  the  dialectic 
to  display  the  inherent  identity  of  the  two  relations.  To 
conceive  (or  name)  the  relations  in  contradistinction  would 
be  to  miss  the  very  truth  revealed  by  the  connotation.  It 
is  everywhere  discovered  that  the  process  of  philosophical 
advance  is  in  great  part  the  identification  of  old  ideas,  long 
in  use  by  the  mind  in  its  experience,  with  ideas  which  to 
brute  or  naive  observation  appear  irrelevant  or  distinct. 
Reflection  upon  the  pure  thought  brings  out  the  implicit 
identity  with  the  category  already  named. 

(3)  In  particular  the  case  of  external  and  internal  di- 
vision in  equal  ratios  is  discovered  to  be  a  harmony  very 
prevalent  in  nature.  Such  division  of  a  sect  is  styled 
"harmonic  division."  (Cy.  any  Geometry  and  any  scien- 
tific treatise  on  physics.) 

(4)  Now  consider  the  two  plane  figures  {A),  a  triangle 
and  any  straight  (cutting  the  triangle  or  not) ;  and  {B),  a 
triangle  and  straights  joining  any  point  (in  or  out  of  the 
triangle)  to  the  vertices  of  the  triangle.  Under  the  ex- 
tended conception  of  the  division  of  a  sect  by  a  point,  the 
straight  in  A  divides  each  side  of  the  triangle ;  and  of  the- 
straights  in  B,  each  divides  the  side  of  the  triangle  oppo- 
site to  the  vertex  through  which  it  passes,  in  such  wise 
that  the  product  of  the  three  ratios  of  the  segments  of  the 


82  NUMBER   AND   ITS  ALGEBRA. 

sides  is  1  (provided  that,  of  adjacent  segments  in  different 
sides,  if  one  be  tlie  antecedent,  then  the  other  sliall  be  the 
conseqnent  of  its  respective  ratio).  Tliis  is  assuredly  a 
most  impressive  exhibition  of  unsuspected  laivfulness  in 
a  fact  seemingly  a  very  type  of  haphazard.  But,  be  it 
noted,  the  inverse  of  neither  A  nor  B  is  true.  Now,  it  is 
an  established  principle  that  when  the  inverse  of  any  prop- 
osition is  not  true,  it  is  because  the  subject  of  the  direct 
statement  has  been  more  closely  limited  than  truth  required. 
It  is  clear  that  the  inverses  of  A  and  B  are  flat  contra- 
dictions. 

But  if  the  ratio  of  sects  from  the  same  point  be  consid- 
ered positive  if  one  sect  is  part  of  the  other,  and  negative 
if  extending  oppositely,  then  the  easily  demonstrated  con- 
clusion of  A  is  that  the  product  of  the  said  ratios  is  -f  1 5 
and  of  B  that  the  product  is  —  1.  The  inverse  of  each 
now  holds ;  that  is,  if  three  points  divide  the  sides  of  a 
triangle  so  that  the  product  of  the  ratios  taken  as  stated  is 
+  1,  then  the  points  are  co-straight ;  and  if  the  product  of 
the  ratios  is  —  1,  then  the  joins  of  the  points  with  the  ver- 
tices concur. 

(5)  The  student  should  fully  realize  what  is  here  as- 
serted ;  and  to  this  end  let  him  draw  a  triangle  and  then 
dash  straights  at  random,  cutting  the  triangle  or  not: 
Every  one  of  them  divides  the  sides  of  the  triangle  in  pre- 
cisely the  same  way ;  and  ifnumher  he  positive  and  negative, 
given  three  points  so  dividing  the  sides,  they  are  co-straight. 
Again,  draw  a  triangle,  dot  at  random  points,  in  or  out 
of  the  triangle :  Any  one  of  these  points  joined  to  the  ver- 
tices gives  straights  which  divide  the  opposite  sides  in 
precisely  the  same  way  ;  and  if  number  be  jjosltlve  and 
negative,  given  this  Avay  of  division  .of  the  sides  by  three 


PRINCIPLE   OF   CONTINUITY.  83 

points,  the  straights  joining  the  point  to  the  vertices  come 
together  in  one  point. 

110.  When  it  is  considered  that  the  preceding  ilhistra- 
tion  recites  merely  one  of  ten  thousand  examples,  number 
is  proved  to  be  positive  and  negative,  —  not,  be  it  under- 
stood, as  a  convention,  but  as  a  necessity  of  thought.  Men 
who  represent  this  qualitative  distinction  as  arbitrary,  or 
as  purely  a  matter  of  algebraic  symbols,  do  not  appreciate 
the  evidence,  or  do  not  understand  what  proof  in  such 
premises  means.  Moreover,  it  must  never  be  overlooked 
that  a  stiir  higher  order  of  proof  is  afforded  in  the  devel- 
opment of  the  pure  idea,  regardless  of  any  adaptations  to 
external  facts.  When  this  or  that  development  of  the 
pure  science  of  number  is  to  lind  application  to  facts  of 
other  sciences  is  a  secondary  matter.     {Cf.  §  117). 

111.  Similar  illustrations  might  be  given  to  show  the 
adaptability  of  number  to  facts  presented  by  nature,  if  the 
other  modes  of  number  resulting  from  the  application  of 
the  Principle  of  Continuity  are  recognized.  Presentation 
of  such  evidence  must  be  postponed  for  the  most  part  to 
subsequent  mathematical  studies ;  and  I  shall  in  this  con- 
nection only  ask  you  to  observe  that  the  Principle  of  Conti- 
nuity, as  enunciated  in  Section  103,  unities  all  the  partial 
explanations  of  number  which  you  will  find  advanced,  or 
implied,  in  various  treatises  ;  and  to  reflect  that  the  man 
who  in  his  own  opinion  discovers  the  entirely  New  is  prob- 
ably on  the  pathway,  not  of  truth,  but  of  estrangement. 
If  his  system  refutes,  in  utter  antagonism,  preceding  sys- 
tems, it  is  likely  to  be  refuted  by  a  successor.  In  all 
philoso})hy  and  science,  advance  has  been  genuine  only  in 
systems  which  have  been  synthetic,  and  unifying  of  pre- 
vious efforts  in  a  harmony  of  thought.     No  development  of 


84  NUMBER    AND   ITS    ALGEBllA. 

thought  must  be  regarded  as  a  disjoined  succession  of  dead 
results,  but  as  living  insights  in  one  line,  each  piercing 
deeper  and  deeper. 

112.  In  this  light,  note  that  the  revelation  in  antiquity 
of  fractional  and  surd  numbers,  and  the  recognition  of 
number  as  positive  and  negative  which  has  prevailed  for 
two  centuries  (these  may  be  regarded  as  the  "  first  "  (§  78) 
and  second  extensions  of  the  number-concept),  are  both 
merely  special  cases  of  the  universal  principle  here  ad- 
vanced. 

113.  To  generalize  is  to  see  in  a  multiplicity  of  objects 
similar  relations  to  one  form  of  mental  activity  that  knows 
those  objects.  But  until  one  sees  the  need  of  a  deeper 
principle  than  that  which  he  has  hitherto  employed,  he 
does  not  seek  a  way  leading  from  what  is  known  to  him  to 
knowledge  beyond.  Any  idea  is  at  first  bare  of  manifold 
essential  relations,  external  and  internal.  By  reflection 
such  relations  are  slowly  revealed.  During  the  process 
the  idea  may  seem  derivative  from  the  relations  {Cf.  geo- 
metric definitions  of  number,  §  25)  ;  but  finally  this  loose- 
ness must  be  reduced  to  order,  and  then  all  its  belongings 
are  seen  to  unfold  from  the  idea  itself,  ■ —  *'  first  the  blade, 
then  the  ear,  after  that  the  full  corn  in  the  ear." 

114.  What  has  just  been  said  would  do  for  a  description 
of  the  famous  dialectic  which  Hegel  describes  as  "the 
self-movement  of  the  notion  (^Begrlff).'^  Indeed,  it  is  not 
much  more  than  a  paraphrase  of  its  description  by  Dr. 
Harris,  "  Seize  an  imperfect  idea  and  it  will  show  up  its 
imperfection  by  leading  to  and  implying  another  idea  as 
a  more  perfect  or  complete  form  of  it.  Its  Imperfection 
tv'dl  slioia  itself  as  dependence  on  another.''''      (Italics  mine.) 

115.  I  know  no  other  method  by  which  the  teacher  can 


ZERO.  85 

lead  a  student  to  attain  for  himself  a  concex^t  of  number 
adequate  to  any  comprehension  of  modern  mathematical 
analysis.  Each  tentative  idea  of  number  must  pass  over 
into  the  next  deeper  as  the  result  of  further  and  further 
insight  into  the  subject. 

It  remains  to  apply  in  characteristic  cases  the  Principle 
of  Continuity,  discovering  from  the  formula  of  definition  of 
any  operation  X^Q  nature  of  the  resulting  number,  as  well 
as  the  efficacy  of  any  such  new  phase  of  number  in  any 
combination  in  the  defined  operations. 

XII.     SiGXIFICAK-CE   A?^D   EfFICACT   OF   NUMERICAL   OPER- 
ATIONS  Under  the   Ultimate   Concept. 

116.  The  very  first  application  of  the  Principle  of  Con- 
tinuity to  the  generalization  of  the  operation  Subtraction, 
displays  a  number  sui  generis,  which  is  of  immense  impor- 
tance in  analysis.  The  formula  of  definition  of  subtrac- 
tion is  (tnde  §  42)  b  —  a  -{-  a  =  b.  Then  a  —  a  =  what 
number  ?  The  formula  declares  that  it  is  a  number  which, 
added  to  a,  makes  a ;  that  is,  it  is  a  number  which  has 
710  efficacy  in  addition,  and  therefore  none  in  subtraction. 
The  best  and  only  unprejudicial  name  for  this  number  is 
zero.  Its  symbol  in  arithmetic  and  in  the  algebra  of 
number  is  0. 

I  trust  that  at  least  it  has  been  made  clear  to  the 
student  that  it  is  only  the  very  primary  and  crudest  con- 
cept of  number  which  would  consider  zero  "  nothing  ; "  for 
although  of  no  efficacy  in  addition  or  subtraction,  it  will 
presently  be  seen  to  exert  extraordinary  effect  in  every 
other  operation.  I  entreat  the  student  not  to  slip  at  this 
point ;    for  the   human    mind,  once    made    sensible  of    its 


86  NUMBER   AND   ITS   ALGEBRA. 

powers,  will  never  afterwards  suffer  its  conception  to  be 
clogged  by  the  tyranny  of  material  categories.  Moreover, 
it  may  quite  commonly  be  found  necessary  to  translate 
into  correct  terms  much  discourse  in  mathematical  trea- 
tises, even  when  written  by  men  eminent  for  skill  and 
learning,  to  say  nothing  of  inadequate  or  erroneous  pre- 
sentations in  works  on  physics  and  applied  mathematics 
in  general.  For  example,  you  may  read  a  Trigonometry 
which  defines  the  trigonometric  ratios  not  as  numbers,  but 
as  sects  (pieces  of  straight  lines)  ;  yet  you  can  often  catch 
the  author  adding  one  of  his  bits  of  straight  lines  to  2 
or  32,  and  in  a  context  where  he  really  means  the  number 
2  or  3^,  etc.  Occasionally  you  will  meet  denial  or  even 
ridicule  of  all  that  I  endeavor  to  lead  you  to  see,  and  per- 
haps by  a  man  of  world-wide  fame.  For  example,  in  a 
didactic  treatise  on  Mathematics  by  De  Morgan,  published 
in  a  serial  Library  of  Useful  Knowledge,  London,  1836, 
zero  is  conceived  to  be  ''  nothing "  ;  for  on  page  23  one 
reads,  "  Above  all,  he  must  reject  the  definition,  still  some- 
times given  of  the  quantity  —  a,  that  it  is  less  than  nothing. 
It  is  astonishing  that  the  human  intellect  should  ever  have 
tolerated  such  an  absurdity  as  the  idea  of  a  quantity  less 
than  nothing ;  above  all,  that  the  notion  should  have  out- 
lived the  belief  in  judicial  astrology  and  the  existence  of 
witches,  either  of  which  is  ten  thousand  times  more  possi- 
ble." The  truly  astonishing  thing  concerning  the  human 
intellect  is  that  such  a  man  as  De  Morgan  could  have  writ- 
ten this  sentence,  familiar  as  he  must  have  been  with 
Newton's  distinction,  '' Quantitates  vel  Aflirmativa^  sunt 
seu  majores  nihilo,  vel  Negativse  seu  nihilo  minores." 
But,  although  deficiency  is  quite  as  quantitative  as  excess, 
the  whole  remark  is  impertinent ;  for  zero  is  not  "  nothing." 


NEGATIVE  NUMBER.  87 

Negative  numbers  are  unquestionably  less  than  zero.  Yet, 
taking  liim  at  his  own  word,  De  Morgan  should  have  hesi- 
tated before  ridiculing  as  crazy  the  careful  dictum  of  as 
powerful  and  piercing  an  intellect  as  has  ever  served  man's 
will. 

117.  Before  investigating  the  efficacy  of  zero  in  other 
operations,  let  us  look  into  further  results  of  the  generali- 
zation of  subtraction. 

What  are  the  properties  of  the  resulting  number  in  the 
operation  b  —  a,  ii  b  <.a?  Consider  the  results  of  the 
following  series  of  operations,  1  +  2;  1  +  1;  1;  1  —  1; 
1_2;  1-3;  1-4,  etc. 

Here  we  have  a  series  of  numbers  which  at  first  decrease 
by  1,  viz.,  3 ;  2 ;  1 ;  0.  The  subsequent  numbers  respec- 
tively answer  the  questions,  what  number  added  to  2 
makes  1,  added  to  3  makes  1,  added  to  4  makes  1  ?  Now, 
in  these  operations  the  sums  remain  the  same,  and  the 
given  summands  in  each  case  increase  by  1 ;  it  is  clear, 
therefore,  that  the  required  summands  must  decrease  by 
1.  Moreover,  these  numbers  in  additive  combination  nul- 
lify 1,  2,  3,  etc. ;  that  is,  make  the  sum  in  each  case  zero. 
Thus,  1  +  (1  -  2)  =  (1  +  1)  -  2  ==  0;  2  +  (1  -  3)  =  (2  + 
1)  _  3  =  0  ;  3  +  (1  -  4)  =  (3  +  1)  -  4  =  0.  Such  reflec- 
tions reveal  an  unending  series  of  discrete  numbers  de- 
creasing from  zero,  each  less  than  the  preceding  by  1. 
Their  effect  in  nullifying  1,  2,  3,  etc.,  in  addition,  renders 
appropriate  the  appellations  ^jostYive  and  negative  to  pri- 
mary numbers  and  these  now  discerned.  These  terms  are 
established  terms  in  logic,  and  are  expressive  of  just  such 
a  relation  of  clean-contradictory  as  has  been  discovered  in 
these  modes  of  number.  On  this  score,  either  might  be 
called  positive  and  the  other  negative ;  but  every  propriety 


88  NUMBER   AND  ITS   ALGEBRA. 

commends  the  course  adopted  —  primary  numbers  are  posi- 
tive, and  such  results  as  we  have  just  considered,  negative. 
That  negative  number  finds  unlimited  corroboration  in 
adaptation  to  the  facts  of  other  sciences,  has  been  amply 
illustrated  (§  109)  ;  but  its  existence  for  pure  mathematics 
is  nowise  dependent  upon  such  circumstances.  Negative 
number  should  never  be  defined  or  explained  by  such  oppo- 
sitions as  right  and  left,  up  and  down,  forivard  and  hack- 
ward,  north  and  south,  past  and  future,  capital  and  debt ; 
but  always  in  its  essential  character  as  number. 

118.  Writing  pos.  for  positive,  and  neg.  for  negative,  it 
is  evident  that  pos.  a  +  neg.  b  =  pos.  a  —  pos.  b  ;  for  pos.  1 

—  pos.  2  =  neg,  1,  therefore,  by  definition  of  subtraction, 
pos.  2  +  neg.  1  =  pos.  1 ;  but  pos.  2  —  pos.  1  =  pos.  1,  etc. 

Also,  since  subtraction  is  the  inverse  of  addition, 

pos.  a  —  neg.  b  —  pos.  a  -f  pos.  h. 

119.  Hereby  Section  42  is  completely  generalized,  and 
the  common  rule  about  "  signs  "  and  parentheses  for  addi- 
tions and  subtractions  established  without  restriction. 

120.  We  are  arrived  now  at  a  matter  of  extreme  impor- 
tance, viz.,  the  dual  significance  of  the  signs  +  and  — . 
One  of  the  most  salient  imperfections  of  ordinary  text- 
books is  their  failure  to  make  a  clear-cut  distinction  be- 
tween the  essentially  double  meaning  of  +,  and  of  — . 
Too  often  the  operational  significance  alone  is  defined, 
although  on  the  next  page  }'ou  may  find  a  complacent 
statement  «  -f  (_  o)  =  0;    whereas,  if  +  means  add,  and 

—  means  subtract,  a  -\-  (—  a)  means,  "starting  with  a,  add 
and  then  subtract  a,"  of  course,  Avith  the  result  a.  And 
under  a  purely  operational  definition  such  an  expression  as 
a  I  —  h  is   like  a  "  sentence  "  made  by  writing  words  on 


DOUBLE   MEANING   OF    +    AND    — .  89 

dice  and  rolling  them  out  of  a  box.  Clifford,  in  liis  zeal 
against  this  abomination,  goes  too  far,  and  gives  three 
totally  distinct  meanings  to  each  of  the  signs.*  His  first 
two  for  each  are  all  that  are  needed  or  justifiable. 

The  names  of  the  signs  are  respectively  "  plus "  and 
"  minus  ;  "  their  meanings  respectively  add  or  positive,  and 
subtract  or  negative. 

It  is,  perhaps,  to  be  regretted  that  beginners  are  not 
taught  to  use  at  first  different  symbols  for  these  wholly 
distinct  thoughts,  and  afterwards  led  to  observe  that  the 
notation  would  be  simplified  if  one  symbol  were  used  in 
both  meanings ;  because  the  context  always  makes  it  clear 
which  is  meant,-  if  the  simple  convention  be  established, 
that,  if  nothing  is  expressed,  ^^ positive  "  is  understood,  and 
if  one  is  omitted,  it  is  the  qualitative,  and  not  the  opera- 
tional, symbol.  Thus,  in  (2)  (-  3),  2  /  -  3,  3--,  V  -  1, 
etc.,  the  meaning  subtract  would  not  make  sense,  and  ambi- 
guity is  impossible  ;  and  in  2  -|-  3  —  4  the  convention  makes 
it  clear  that  the  meaning  is  pos.  2  -\-  pos.  3  —  pos.  4.  It  is 
true  that  2  -f  3  —  4  =  pos.  2  +  pos.  3  -f  neg.  4,  and  although 
less  consistent  than  the  notational  convention  I  recite,  the 
expression  might  be  understood  in  this  sense  ;  for  the  result, 
as  we  have  seen  in  Section  118,  is  indifferent.  But  see 
clearly  that  the  sign  cannot  have  both  meanings  at  one 
time ;  for  7  —  9  =  pos.  7  —  pos.  9  =  pos.  7  -\-  neg.  9  =  neg. 
2,  whereas  pos.  7  —  neg.  9  =  pos.  16. 

Kote,  as  in  accordance  with  the  convention  stated,  that  in 
solving  a  synthetic  equation  for  an  unknoAvn  number,  its 
qualitative  nature  is  unknown,  and  no  sign  is  to  be  under- 
stood after  the  sign  meaning  add  or  subtract. 

*  Common  Sense  of  the  Exact  Sciences,  p.  34  et  seq. 


90  NUMBER   AND   ITS   ALGEBRA. 

If  for  any  purpose  it  is  desirable  to  be  quite  explicit,  the 
qualitative  sign  may  be  put  in  parentheses  with  the  number- 
symbol,  with  the  operational  sign  preceding.  Parentheses 
would  hardly  be  used  for  the  first  term,  or  for  a  term  stand- 
ing alone  ;  e.g.,  +  7  -  (+  8)  +  (+  G)  -  (+  9)  =  -  4,  is 
the  full  expression  of  what  is  meant  by  7—  8  -}-  G  —  9  =  —  4. 
Of  course,  if  occasion  rose,  write  -|-7-l-(—  8)  —  (  —  6)  — 
(+  9)  =  —  4.  In  short,  write  Avliat  you  mean,  if  you 
express  fully,  but  remember  that  abbreviations  must  be 
doubly  conventional.      {Vide  §  162.) 

121.  What  is  the  product  of  (—  a)  (+  h)  ? 
Consider  {-(-  m  —  ( -{-  «)}  (-}-  b)  where  vi  >  a. 

By  the  distributive  law,  this  equals  +  bm  —  (+  ba)  ;  but 
by  Section  118  it  equals  {+  m  +(  —  a)}  (  +  /');  but  {-{-  m 

+  (—  <0}  (+  ^0  =  +  ^"^  +(—  ")  (+  f')  by  distributive  law ; 
therefore,  since  -\-  bm  —  ( -\-  ba)  =  -\-  bm  -\-  (~  ba),  -\-  bm 
-j-(_  ba)=  -f  bm+(—  a)  (+  b)  ;  therefore  (—  a)  (+  b)  = 

—  ba. 

Hence  the  common  rule  of  signs. 

122.  What  is  the  product  of  (—  a)  (—  b)  ? 

Consider  {+  m  —  (-f  a)}  (—  b).  Distributing  and  ap- 
plying  Section    121    gives   (4-  m)  (—  b)  —  (-\-  a)  (—  b)  = 

—  bm  —  {—  ba)  =  —  bm  -\-  (+  ba)  ;  but  by  Section  118, 
{+  m  -  (+  a)}  (-  b)  =  {+  m+  (-  a)}  (-  b)  =  -  bm  + 
{—  a)  {—  b),  therefore  (^—  a)  {— b)  =  -[- ba  =  +  ab. 

Hence  the  common  rule. 

123.  Division's  definition  as  the  inverse  of  multiplica- 
tion, of  course,  establishes  the  rule  of  signs  for  division. 

124.  Sections  121,  122,  and  123  render  complete  under 
the  common  "■  rule  of  signs  "  the  freedom  of  distribution 
and  commutation  referred  to  in  Sections  54,  55,  and  56. 

125.  We  are  now  prepared  to  investigate  still  further 


OPERATIONS  UNDER  FINAL  CONCEPT.       91 

the  properties  of  zero.  We  have  seen  that  it  has  no  effi- 
cacy in  combination  with  other  numbers  in  addition  and  in 
subtraction.  What  is  its  efficacy  in  multiplication,  in 
division,  in  involution,  in  evolution,  and  in  finding  the 
logarithm  ? 

126.  ^Vhat  is  the  product  (a)  (0)  ? 

Consider  ba  —  ba  =  0.  By  distributive  and  commutative 
laws,  and  Section  122,  ba  —  ba  =  (b  —  b)  (+  «)  =  (+  a) 
(b  -  b)  =  {b  -  ?y)  (-  o)  =  (-  a)  (b  -  b);     whence    (0) 

(+  a)  =  (+  a)  (0)  =  (0)  (-  «)  =  (-  ")  (P)  =  0'  o^'  ^^^^^y> 
regardless  of  positive  or  negative  quality  of  a, 

(«)  (<^)  =  (0)  («)  =  0. 

127.  Note  that  as  an  independent  number  zero  is  with- 
out qualitive  distinction  of  positive  and  negative ;  for 
a  4-  0  =  a  —  0,  hence  -f-  0  =  —  0. 

128.  It  may  be  a  profitable  comparison  to  call  atten- 
tion expressly  to  a  unique  property  of  1  in  multiplication 
and  division  ;  thus,  — 

(r?)  (-(-  1)  =  rt,  and  a  /  -{-  1  =  a,  that  is  to  say,  x  (+  1)  = 

-  (+  !)• 

Also  (a)  (—  1)  =  —  a,  and  o  /  —  1  =  —  a,  i.e.,  X  (—  1) 

=  -  (-  !)• 

129.  0  X  0  =  0,  for  (a  _  a)  (Z»  -  ?;)  =  0  X  0  =  ab  —  ah 

_  ah  +  aJ>  =  0. 

130.  r.ut  what  is  the  result  0 /O  ? 

This  case  is  of  extreme  importance.  Failure  to  compre- 
hend it  when  it  comes  into  systematic  use  in  the  Calculus 
has  put  a  veil  of  irrational  mystery  over  that  whole  dis- 
cipline. Thousands  of  students,  although  they  have  met 
and  slightly  used  this  indeterminate  form  before,  yet  inas- 
much as  they  have  regarded  it  a  matter  of   special  con- 


92  NUMBER   AND   ITS   ALGEBRA. 

vention  that  0/0  should  represent  any  number,  are 
dumfounded  to  find  a  discipline  where  a  number  is,  they 
say,  made  zero  in  one  member  of  an  equation,  and  some- 
thing else  in  the  other.  After  a  pathetic  struggle  to  see 
reason  in  their  procedure,  they  commonly  give  over,  and 
accept  the  outrageous  extravagance  that  a  concatenation  of 
deductions  to  be  valid  need  not  have  meaning  in  every 
link ;  that  a  compulsory  conclusion  of  an  argument  does 
not  require  intelligibility  of  its  several  steps;  or  that 
results  may  be  thoroughly  made  out  true  for  reasons  no- 
wise understood. 

131.  "When  the  ratio  0/0  is  first  presented  for  consider- 
ation, one  may  be  disposed  to  jump  to  the  decision  that 
0/0  =  0  or  0/0  =  1;  but  it  is  clear,  from  the  definition 
of  division,  that  in  the  synthetic  equation  0/0  =  y,  any 
number  (0,  3/5,  1,  V2,  2,  3,  etc.)  substituted  for  y  will 
make  a  formula  (§  40),  an  identity.  That  is  to  say,  0  /  0  = 
anv  number. 

V 

Indeed,  this  statement  is  merely  another  way  of  saying, 
"any  number  multiplied  by  zero  gives  zero,"  which  is  com- 
monly accepted  without  objection.  And  both  of  these 
statements  are  only  particular  applications  of  the  postulate 
expressed  in  the  Principle  of  Continuity. 

The  ratio  0/0,  then,  may  be  any  number;  but  in  par- 
ticular instances  it  is  often  a  number  which  may  be  deter- 
mined by  independent  considerations. 

132.  If  two  numbers  (or  any  other  two  magnitudes  of 
the  same  kind)  vary,  their  ratio  varies  ;  but  the  ratio  at 
any  assigned  limits  of  the  variables  is  the  same  as  at  values 
of  the  variables  only  infinitesimally  (vide  §  222)  removed 
from  such  limits.  In  fact,  the  original  definition  of  equality 
of  ratios  contains  this  doctrine.  •    (  Vide  §  83  (6).) 


OPEKATIONS   UNDEll   FINAL   CONCEPT.  93 

Consider  the  functions  of  x,  x^  —  1,  and  x  —  1. 

3.2  _  1 
What  is  the  ratio wlien  x  =  1? 

X  —  1 

The  ratio  may  be  evahiated  Avithout  hesitation,  as  x 
assumes  various  values,  until  a;  =  1  is  reached,  when  both 
functions  vanish,  and  the  ratio  assumes  the  indeterminate 
form,  0/0.     But  when  x  differed  only  infinitesimally  from 

1,  beyond   objection,  =  x  -\-  1,  which   differs   only 

X  —  1 

infinitesimally  from  2.     Therefore,  when  x  =  1,  the  ratio 

-  =       ~ —  =  a;  +  1  =  2,  absolutely, 
0        x-1  ^  '  -^ 

There  is  no  trickery  here.  The  Calculus,  with  its 
astonishingly  powerful  algorithm,  applies  such  numerical 
interpretations  to  concrete  magnitudes ;  nor  would  it,  in 
my  opinion,  be  out  of  place  in  this  connection  to  give  illus- 
trations of  the  wonderful  propriety,  and  accordance  with 
independent  facts,  of  this  method,  but  out  of  deference  to 
established  custom  —  usus  tyranmis  —  I  leave  such  corrobo- 
rations to  future  studies,  with  the  simple  assurance  of 
their  cogency.  I  shall  only  set  forth  one  more  very  simple 
illustration  of  an.  evaluation  of  a  ratio  0/0.  Consider 
the  following  two  functions  of  y,  2  y  -|-  3  y-  +  4  y^,  and 
3  y  -\-  ^  y"^  -\-  21  y^.  Their  ratio  would  be  easily  evaluated 
for  particular  finite  values  of  y ;  but  suppose  the  variable 
y  becomes  zero,  what  then  is  the  ratio  of  the  functions  ? 

If  y  =  Q,       li^L+ll!  +  li^  =  0/0. 

And  if  each  term  be  divided  by  y  we  have 

2  +  3.v-f-4.v^   _  2  /3,  when  y  =  0. 


94  NUMBER    AND    ITS   ALGEBRA. 

Now,  it  might  well  be  objected  that  in  dividing  by  y,  if 
y  =  0,  the  most  we  could  say  would  be 

2(y/y)  +  3yO////)+4rO////) 

3(y/z/)  +  9y(y/y)  +  27y^(y/y)' 

and  that  this  remains  as  obscure  as  the  original  if  0  /  0 
is  any  number.  The  explanation  is,  that  the  ratio  of  one 
and  the  same  variable  to  itself  is  constantly  1 ;  a  J  a  =  1 
always.     Therefore,  even  when  y  ==  0,  y  /  y  =  1,  and  if  so, 

^(y/y)+3y(y/y)+4y^(y/y)  _  2  +  3y  +  4y^  _^ 

It  would  take  us  too  far  afield  to  go  further  into  the  doc- 
trine of  limits  of  variable  magnitudes  and  infinitesimals, 
and  the  appropriate  application  of  number.  The  whole 
question  of  the  use  of  this  indeterminate  form  0/0  may 
not  improperly  be  postponed  by  the  student,  who  for  the 
present  might  content  himself  with  the  discernment  that, 
whether  it  be  possible  to  evaluate  0  /  0  or  not  in  particular 
problems,  0/0  may  be  any  number. 

133.  What  is  the  result  of  the  operation  0  /  ct  ?  and 
what  of  ffl  /  0  ? 

The  first  asks  the  question,  what  number  multiplied  by 
a  gives  zero ;  and  from  the  formula  of  definition  and  Section 
126,  the  answer  is  evidently  zero. 

Also  0/  +  a  =  0  =  0/-a. 

The  second  asks  the  question,  what  number  multiplied 
by  zero  gives  a  ? 

Erom  Section  126  it  is  evident  that  no  number  3-et  dis- 
cerned answers  this  question. 

But  a  consideration  of  the  continuously  increasing  ratios 
(vide  §§  81,  82)  of  the  same  number  to  a  decreasing  series 


INFINITY.  95 

of  numbers,  reveals  that,  if  the  ratio  -)-  a  /  0  is  a  number, 
it  is  one  greater  than  any  primary  number,  and  of  peculiar 
"efficacy  in  operational  combination.  This  number,  whose 
reality  is  requisite  for  untrammelled  numerical  analysis, 
is   called  2^ositive  infiiiity,  and  notationally  expressed  as 

-|-  CO  - 

Similar  generalization  under  the  Principle  of  Continuity 
makes  —  a  /  0  negative  infinity,  written  -co  . 

134.  The  discovery  of  many  properties  of  infinity,  posi- 
tive and  negative,  must  be  left  to  future  studies ;  as  well 
as  the  principles  of  evaluation  of  ratios  of  infinities  dif- 
ferently derived,  analogous  to  evaluations  of  ratios  0/0. 
{Vide  §  132.) 

It  Avill  be  found  that  in  the  application  of  ISTumber  to  cer- 
tain magnitudes  (e.g.,  straight  lines  in  Euclidean  Geome- 
try) that  for  them  it  appears  the  points  at  infinity  coincide. 
Other  "  one-dimensional "  (vide  §  232)  magnitudes  show 
a  double  absolute  :  for  example,  ProhahiUty  ranges  from 
absolute  certainty /or,  to  absolute  certainty  against. 

Without  going  too  deeply  into  philosophical  questions, 
it  may  be  remarked  that  Hegel,  in  discussing  the  mathe- 
matical infinite,  "  points  out  that  the  mathematical  infinite 
.  .  .  uses  the  idea  of  the  true  infinite,  and  therefore  stands 
higher  than  the  so-called  metaphysical  infinite.  The  latter 
opposes  the  infinite  to  the  finite  as  the  mere  negative  of 
the  latter,  and  thereby  makes  two  finites,  the  former  the 
void  of  the  latter  ;  whereas  the  mathematical  infinite  ex- 
presses self-relation  as  its  true  form."*  Much  might  be 
said  also  of  how  important  to  philosophy  is  the  mathe- 
matical concept  of  continuity.     Indeed,  many  of  Hegel's 

*  HegeVs  Logic,  Harris,  !>.  278. 


96  NUMBER    AND    ITS    ALGEBRA. 

conceptions  are  true  only  as  glimmerings  of  wliat  mathe- 
maticians had  before  made  clear,  or  have  since  illuminated. 

135.  I  present  in  tabular  form  *  the  possible  meanings' 
of  the  ratio  x  j  ij,  as  x  and  y  independently  vary  from  0 
to  ao  .  The  student  can  readily  verify  the  statements,  and 
extend  them  to  cover  distinctions  of  positive  and  negative 
in  X  and  y  :  — 

(  1  )  X  I  y'\'&  finite  if  x  is  finite  and  y  finite. 

(  2  )  may  be  finite  if  a;  =  0  and  y  =  0, 

(  3  )  or  if  a;  ^=  CO  and  v/  =  cc  . 

(4)a:'/?/  =  0  ifa;  =  0  and  y  not  0, 

(  5  )  or  if  .T  not  co   and  y  ^  c/j  . 

(  6  )  may  =0  if  a;  =  0  and  y  =  0, 

(  7  )  or  if  a-  =  oo  and  y  =  cc  . 

(8)  x  /  y  =  ao  ifa-=co   and  y  not  cci , 

(  9  )  or  if  a;  not  0  and  y  =  0. 

(10)  may  =  oo        if  a;  =  0  and  y  =  0, 

(11)  or  if  a;  =  CO  and  ?/  =  oo  . 

136.  Of  course  oo  +  oo  =  co  .  But  oo  —  oo  is  indeter- 
minate ;  since  any  number  (0,  finite,  or  infinite)  substi- 
tuted for  X  satisfies  the  synthetic  equation  oo  +  a;  =  oo  . 

137.  Of  course  oo  X  co  =  oo .  But  OXco  =  coXO  is 
indeterminate ;  since  the  multiplications  of  which  Section 
135  (5),  (7)  are  the  inverses,  show  0  X  oo  =   any  number. 

138.  Various  considerations  dependent  iq^on  the  con- 
tinuity of  number  confirm  the  interpretation  that  a;"  =  1, 
if  X  is  finite.  But  it  may  suffice  to  consider  that  if  x,  y, 
and  z  are  finite,  a;^  -=-  a;^  =  x^-' ;  and  it  y  =  z,  1  =  x'J  -i- 

xy  =  xy-y  =  x". 

*  A  similar  table  occurs  in  Chrystal's  Text  Book  of  Algebra,  Part  I., 
p.  317. 


OPERATIONS    UNDER   FINAL   CONCEPT.  97 

139.  Evidently  (1)      0-^  =   0  if  a:  is  finite. 

(2)  a;+*  =00  if  ic  is  finite  and  >  1. 

(3)  =   0  if  rr  <  1  and  >  0. 

(4)  x"^  =   0  if  33  is  finite  and  >  1. 

(5)  =  CO  if  ic  <  1  and  >  0. 

(6)  0+-  =0.         (8)  a;  +=»  =  00  . 

(7)  0-=  =  00  .       (9)  ^  -="  =  0. 

As  the  student  may  convince  himself.    (§  143  is  anticipated.) 

140.  But  the  results  (1)  0°,  (2)  ^  °,  (3)  1+^,  (4)  l-«  are 
indeterminate ;  as  may  be  seen  most  readily  by  considering 
that  x^  =  m2'^°^ra-^,  where  m  is  finite  and  greater  than  -|-  1. 
x^'  is  accordingly  determinate  when  y  log„j  x  is  determinate, 
and  indeterminate  when  ylog^a;  is  indeterminate.  The 
cases  when  ?/log,„«  is  indeterminate  are,  by  Section  137  :  — 

(1)  When  y  =  0,  log,„  x  =  -co";  i.e.,  when  y  =  0,  cc  =  0. 

(2)  When  y  =  0,  log„j,x  =  -[-  co  ;  i.e.,  when  y  =  0,  .r  =  oo  . 

(3)  When  ?/  =  -j-  oo  ,  log,„a3  =  0  ;  i.e.,  when y  =  -j-c/^  ,x  ^  1. 

(4)  When  y  =  —  cc  ,  log,„ic  =  0  ;  i.e.,  when  y  =  —  ct)  ,x  =  1. 

141.  Every  indeterminate  form  may  be  reduced  to  0/0, 
and  in  this  sense  it  may  be  said  that  the  one  fundamental 
case  of  indetermination  is  0/0.     For  example :  — 

00-^  =  1/0-1/0  =  1^  =  0/0; 
^/^  =  ^  =  0/0. 

142.  Let  the  student  tabulate  from  the  foregoing  sec- 
tions all  the  indeterminate  operations. 

He  must  be  content  to  postpone  investigation  into  the 
evaluation  of   these  indeterminate    results    as   they  arise 


98  NUMBER   AND   ITS   ALGEBRA. 

from  particular  functions  of  variables,  regarding  Section 
132  as  a  simple  example  of  the  general  principle. 

143.  It  remains  to  investigate  the  efficacy,  as  exponents 
of  evolution,  of  fractional,  surd,  and  negative  numbers. 

What  is  the  meaning  of  the  operation  a^,  if  a  is  positive 
and  finite,  and  x  a  fraction  ? 

The  conclusion  is  corroborated  by  the  continuity  of  num- 
ber, by  countless  correspondences,  and  by  perfect  consist- 
ency with  all  other  laws  ;  but  regarding  the  Law  of  Indices 
as  the  essential  definition  of  the  operation,  the  meaning  is 
immediately  revealed.  Thus  :  let  a'" ' ",  where  7n  and  n  are 
positive  integers,  equal  z.  Then,  since  «"«/«  is  subject  to 
the  Law  of  Indices,  z^  =  zzz  .  .  .  n  factors  =  a"'^"a'"^" 

.    .    ,    n    factors  =  «"-/"+«/»    ■    ■    ■    n  terms  ^  „m_        rj^^r^^    -^    ^^ 

say,  z  is  a.  number  whose  nth  power  is  a'";  or  z  is  an  «th 
root  of  a"';  i.e.,  a""'"  =  V«'". 

In  particular,  if  m  =  1,  ct} '  ^  =  V«. 

As  we  saw  in  Section  94,  the  operation  ■>/«  (where  a  is 
positive)  is  always  possible,  in  the  sense  that,  if  the  result 
be  a  surd  number,  it  can  be  determined  to  any  degree  of 
approximation.     (But  see  §  153,  et  seq.) 

144.  It  will  appear  in  the  studies  to  which  these  lec- 
tures are  introductory  that  there  are  n  nth.  roots  of  a, 
where  n.  is  a  primary  number  ;  but  the  student  may  observe 
now,  that  when  n  is  even  there  are  at  least  two  roots  of 
a,  one  the  negative  of  the  other;  e.g.,  4}/^=  J^  2.  But 
note  that  the  law  of  indices  has  regard  only  to  the  corre- 
sponding roots  of  numbers,  simply  because  V«  Va  does 
not  equal  a,  if  one  positive  and  one  negative  root  be  taken. 
(ride%^  191  and  146.) 

145.  It  is  necessary  to  say  at  this  point  that  we  must 
either  use  the  terms  "  rational/'  "  irrational,"  ''  real,"  and 


TERMINOLOGY.  99 

"  imaginary,"  or  invent  equivalents.  ( Vide  Introduction, 
p.  16,  and  §  101.)  The  terms  are  unquestionably  abusive, 
and  perhaps  the  time  is  ripe  for  a  protest.  No  number  is 
irrational,  and  all  numbers  are  real.  Therefore,  if  merely 
as  an  experiment,  I  shall  be  consistent  in  calling  numbers 
commensurable  (tvith  1,  understood)  where  the  text-books 
say  "rational ;  "  either  incommensurable  or  surd,  where  they 
say  "irrational;"  radical-surd,  where  they  say  ''surd" 
(^Cf.  §  83)  ;  protomonic,  where  they  say  "real"  ;  neoinonic, 
where  they  say  "  imaginary  "  (unless  they  say  "  imaginary  " 
when  they  mean  complex)  ;  and  when  functions  or  opera- 
tions are  spoken  of  as  "rational"  or  "irrational,"  in  sub- 
stituting stlrpal  and  radical  respectively.  These  words, 
except  2}'>'otomonic  and  sthpal,  are  in  good  usage  either 
exactly  or  approximately  in  the  senses  defined.  Proto- 
monic  and  sthyal  I  coin ;  reluctantly,  but  unavoidably.  I 
hope  they  justify  themselves  as  antitheses  of  neomonic  and 
radical.  Of  course,  surd  is  not  much  better  etymologically 
than  "  irrational ;"  but  the  metaphor  is  dead,  and.  conse- 
quently harmless.  Concerning  commensurable,  see  Section 
205.     (See  also  §  156.) 

146.  Before  passing  to  other  cases  of  the  exponential 
function  a^,  it  is  proper  to  call  attention  to  certain  para- 
doxes which  may  arise  in  the  interpretation  of  such 
functions.  (Cy*.  §  191.)  For  example,  a*  ^-=  a'^.  But  as  a 
fractional  index,  a^  ''^  means  Va*  =  i  a'^  which  at  first  sight 
might  seem  to  assert  that  a"  =  -^  a~.  Likewise,  one  might 
be  led  to  say,  since  (a'")"  =  a"">  =  (a")"',  (-i^'-y  =  (■i-y^, 
and  so  (-t-  2)^  =  J-  4,  that  is,  +  4  =  ^  4.     (Cf.  §  144.) 

Such  difficulties  will  arise  in  a"'^",  etc.,  when  mfn  is 
not  in  its  lowest  terms,  a*'^  =  a^  is  not  a  radical  function 
at  all;   though  it  is  quite  true  that  the   second  roots  of 


100  NUMBER    AND    ITS    ALGEBRA. 

a*  are  +  a^  and  —  a^.  The  law  of  indices  is  not  a  matter 
of  arbitrary  or  meaningless  symbols,  but  of  facts.  If 
algebraic  expressions  are  not  regarded  as  logical  state- 
ments, and  full  account  taken  of  the  nature  of  the 
derivation  of  one  equation  from  another,  apparent  con- 
tradictions will  often  arise.     (^Cf.  §  319  et.  seq.) 

147.  What  is  the  effect  of  a  negative  exponent  of  invo- 
lution ? 

Consider  «-"»  =  a"'"  X  a™-  j  a"^. 

By  law  of  indices, 

a-'"  X  a'''  I  a'''  =  a- '"  +  '"'  -r-  a'"  =  a"  /  a'"  =  1  /a™, 
by  Section  138  ;  therefore, 

«-"»  =  1  /a"". 

That  is  to  say,  a~'"  is  the  reciprocal  of  a"^. 

148.  The  continuity  of  number  at  once  extends  all  that 
has  been  shown  to  be  true  of  integral  and  fractional  ex- 
ponents to  surd  exponents. 

Thus  in  the  function  a^,  whether  x  be  commensurable  or 

surd,  we  can  always  find  two  fractions,  m  /  n  and  , 

between  which  x  lies,  and  which  differ  by  as  little  as  we 
please.  As  stated  in  Section  95,  in  calculation  we  use 
a^jn  instead  of  a^,  where  m/nis  a  fraction  closely  approxi- 
mating the  surd  x. 

149.  When  a  is  positive  and  >  1,  and  regarding  only 
protomonic  positive  roots,  a^  is  a  continuous  function  of  x, 
passing  through  all  values  from  0  to  -(-  co  ,  as  x  varies  from 
—  oo  to  -f  CO  .     Thus,  — 

a^  is         0,  <  1,  1  /«,  1,  >  1,       a,  +  00 

when  X    is  —  oo  ,     —  ,  —  1,  0,     -\-,  -]^  1,  -]-  cc  . 


LOGARITHMS.  101 

When  a  is  positive  and  <  1,  the  vahies  of  «^  are  +  co  ,  >  1 
1  /  a,  1,  <  1,  a,  0,  corresponding  to  the  same  values  of  x. 

150.  As  has  been  explained,  b  =  a^  and  x  =  log„  b,  are 
merely  different  ways  of  writing  the  same  functional  rela- 
tion. Thus  all  laws  and  properties  of  logarithms  are  de- 
rivable from  the  principles  of  involution,  in  brief,  from  the 
law  of  indices.  Until  the  uses  of  logarithms  and  the  con- 
struction of  logarithmic  tables  are  investigated,  it  is  enough 
to  say  that  for  the  same  base  the  following  are  the  leading 
properties  of  logarithms,  —  as  the  student  may  easily  dis- 
cover from  the  law  of  indices  :  — 

(1)  The  log.  of  a  product  of  positive  numbers  is  the 
sum  of  the  logs,  of  the  factors. 

(2)  The  log.  of  the  quotient  (ratio)  of  two  positive 
numbers  is  the  log.  of  dividend  minus  log.  of  divisor. 

(3)  The  log.  of  any  power  of  a  positive  number  is  the 
log.  of  the  number  multiplied  by  the  exponent.  (Power  is 
used  in  the  general  sense ;  for  the  statement  is  true  for  all 
exponents,  and  therefore  inclusive  of  the  commonly  sepa- 
rated rule  for  roots.) 

(4)  (log,,  b)  (logs  a)  =  1,  and   log„m  =  -51^  . 

The  base  of  "  common  "  logarithms  for  piirposes  of  final 
calculation  is  10  ;  but  the  base  discovered  to  be  primarily 
appropriate  to  mathematical  investigations  is  an  incommen- 
surable number,  called  e. 

e  =  l-ul  +  —  4-  —  H \-  to  CO  terms  * 

=  2-  7182818284  +. 


*  1  X  2  may  be  abbreviated  2 !         -p^^^^  „  factorial  two," 

1X2X3  may  be  abbreviated  3!      ..  factorial  three,"  etc. 

1X2x3X4  may  be  abbreviated  4 ! 


102  NUMBER   AND   ITS   ALGEBRA. 

The  base  10  gives  logarithms  vastly  more  convenient  in 
calculation  ;  the  base  e,  in  analysis. 

Formulae  (4)  yield  the  simple  process  for  deducing  from 
a  given  table  of  logarithms  to  any  base,  the  logarithms  to 
any  other  base.  Thus,  to  deduce  log„  m  from  logj  m,  mul- 
tiply by  . 

logs  a  ^ 

The  constant  multiplier,  ^j ,  is  called  the  modulus  of 

the  system   whose  base  is  a  with  respect  to  the  system 
whose  base  is  b. 

The  modulus  of  the  system  whose  base  is  10  with  respect 

to  the  system  whose  base  is  e,  is  :; —  =  0-4342944819  +. 

^  loff.lO 


■'a  e 


The  modulus  of  the  system,  base  e,  with  respect  to  the 

common  system,  is  =  2-7182818284  -\-. 

-I      logio  e 

Of  course,  =  logio  ^ ;  that  is  to  say,  the  recipro- 

log.lO 
cal  moduli  of  two  systems  are  reciprocals  in  the  numerical 

sense. 

151.  For  interesting  historical  sketches,  the  student  is 
referred  to  the  articles,  "  Logarithm's  "  and  "  John  iSTapier  of 
Merchiston,"  by  J.  W.  L.  Glaisher,  in  the  Encydopcedia  Bri- 
tannica,  ninth  edition.  A  perusal  of  these  monographs  will 
lead  him  to  appreciate  the  brilliancy  of  Napier's  invention, 
and  the  merit  of  Briggs  and  Vlacq,  as  well  as  the  claims  of 
Byrgius,  a  Swiss  contemporary  of  Napier,  as  an  indepen- 
dent but  crude  inventor.  He  should  bear  in  mind  that  this 
achievement  came  prior  to  the  exponential  notation,  or  any 
clear  idea  among  mathematicians  of  exponential  functions. 
An  attempt  to  prove  —  to  say  nothing  of  discovering  —  the 
laws  of  logarithms,  after  divesting  one's  self  of  knowledge 
of  the  generalization  of  involution  and  all  moderji  advan- 


INADEQUACY   OF   PROTOMONIC   NUMBER.  103 

tages  from  the  correspondences  of  two  series  of  numbers, 
one  in  "  arithmetic  "  and  the  other  in  "  geometric  "  progres- 
sion, would  alford  a  very  high  estimate  of  ISTapier's  genius 
and  acumen. 

152.  The  student  can  easily  discern  tlie  laws  of  the 
function  a^  if  a  is  negative,  and  x  zero  or  integral.  If  x 
is  zero  or  a  multiple  of  2,  the  power  is  positive ;  if  x  is 
odd,  the  power  is  negative. 

153.  Also,  if  a  is  negative,  and  x  fractional,  with  odd 
denominator,  the  power  is  protomonic  (vide  §  145).  In 
other  words,  if  a  is  negative  and  n  odd,  there  is  always 
a  protomonic  nth.  root  of  a.  For  consider  the  function  y", 
where  n  is  an  odd  positive  integer.  The  function  passes 
through  all  values  from  -co  to  0,  as  ?/  passes  from  —  co 
to  0.  Therefore,  there  must  be  some  protomonic  negative 
number,  y,  for  any  negative  nvipiber,  a,  such  that  y^  =  a. 
That  is  to  say,  there  is  always  a  protomonic  odd  root  of 
a  negative  protomonic  number. 

154.  But  if  in  the  operation  a-^,  a  is  negative  and  x  a 
fraction  in  its  lowest  terms,  with  even  denominator,  there 
is  no  result  whatever,  nor  is  the  operation  intelligible  with- 
in protomonic  number.  For  the  function  y",  where  n  is 
an  even  integer,  is  always  positive.  Therefore  there  is  no 
protomonic  number,  y,  such  that  ?/"  is  negative  when  n  is 
even.     That  is  to  say,  there  is  no  protomonic  even  root  of 

4 

a  negative  protomonic  number.* 

155.  Evidently,  then,  unless  the  Principle  of  Continuity 

*  Also  in  terms  of  protomonic  number  there  is  no  logarithm  of  a 
negative  number  to  a  positive  base.  At  this  stage  we  cannot  investigate 
such  functions ;  but  log  (+  a)  ( —  b)  has  been  shown  to  be  indeterminately 
any  member  of  an  infinite  series  of  complex  numbers.  Thus  in  no  case 
are  we  led  out  of  complex  number  as  the  ultimate  generalization. 
(Cf.  §  202.) 


104  NUMBER   AND   ITS    ALGEBRA. 

shall  widen  our  concept  of  number,  the  generality  of 
numerical  operations  abruptly  fails  at  this  point.  But  the 
Principle  of  Continuity  does  apply  here  as  everywhere 
else ;  and  the  power  of  analysis  is  enhanced,  and  the  appli- 
cability of  Number  to  the  relations  of  concrete  magnitudes 
perfected  beyond  the  dreams  of  mathematical  science  prior 
to  this  development. 

Before  taking  this  step,  however,  we  must  investigate  a 
few  fundamental  properties  of  radical-surds. 

156.  In  all  algebraic  expression  of  number  the  student 
must  avoid  confusion  on  account  of  any  possible  value  of  a 
function  for  particular  numbers  in  place  of  the  algebraic 
symbols.  Thus  {j^^'^)""  is  a  radical  function  of  2>  in  alge- 
braic form ;  although,  of  course,  in  cases  where  m  =  2n, 
and  n  an  integer  (p^ '  "^y  =  j)^,  the  nature  of  which  again 
depends  on  the  character  of  p.  Or,  the  -y/x  is  algebrai- 
cally a  radical-surd ;  although  if  a?  =  4,  it  is  commensura- 
ble, and  so  forth. 

It  is  not  necessary  to  be  constantly  "  providing  "  obvious 
conditions.  Intelligent  attention  will  always  secure  com- 
prehension of  the  algebraic  statements  in  the  sense  in- 
tended, whenever  explicit  provision  is  omitted. 

157.  A  radical-surd  number,  or  multiple,  or  fraction 
thereof,  is  called  a  simple,  monomial  radical-surd.  The 
suD^  or  difference  of  two  such,  or  of  one  such  and  a  com- 
mensurate number,  is  called  a  simple  binomial  radical-surd. 

It  will  be  seen  that  every  stirpal  function  of  a  radical- 
surd  can  be  expressed  as  a  simple  radical-surd. 

Two  radical-surds  are  called  similar  when  they  can  be 
expressed  as  multiples  or  fractions  of  the  same  radical- 
surd :  e.g.,  V3/4  and  VlS  are  similar;  for  V3/4  = 
1/2  V3,  and  Vl2  =  2  V3. 


EADICAL-SURDS.  105 

E-adical-surcIs  with  the  same  base  and  same  root-index 
are  called  equiradical ;  e.g.,  a^'^,  a"^,  a"'^. 

Radical-surds  with  the  same  root-index  are  called  of  the 
same  order  —  quadratic,  cubic,  biquadratic,  quintic,  .  .  . 
n-tic;  e.g.,  V3,  5^'",  x"'^  are  quadratic  surds;  V3,  5^^^, 
x"'^  are  quintic  surds. 

158.  From  the  Law  of  Indices  (a'»  a"  =  a"' + ")  it  is 
easily  proved  for  protomonic  numbers  (but  see  §  191)  that 

a"' a"''  =  (aa.y^,  or  (('"■  [>'"■  =  (ab)"K     Thus,  if  m  is  integral 
f^m  ^m  _  ^^^(j  .  .  .  m  factors)  X  (phb  .  .  .  m  factors)  by 
definition, 
=  {ab.ah.ab  .  .  .  m  factors)  by  laws  of  association 

and  commutation, 
=  (a^)™  by  definition. 

And  if  m  is  fractional,  say  ?/i.  =  1  /  n  where  w  is  a  posi- 
tive integer,  («i'"a"''a^"'  .  .  .  n  factors)  X  (^^ '"  ^^ '"^' ^" 
.  .  .  n  factors)  =  (a^'^'h^"')  (a^'^b^'")  .  .  .  n  factors; 
but  the  left-hand  member  equals  ab;  therefore  (a^'"i^'") 
(a^^"6^'")  .  .  .  n  factors  =  ab,  therefore  a^'^'b^'"  (aby'\ 
if  positive  roots  of  a,  b,  and  ab  are  alone  considered  (vide 
§§  144,  146).  _ 

159.  A  special  case,  -\/a"b  =  a^b,  is  important  in  re- 
ducing radical-surds  to  similarity. 

160.  Note  also  ^a  =  VaP ;  for  aV»  =  «"/"?  =  V«p. 

pn  +  q 

161.  Also,  ^(1^^  +  "  =  aP^ai;iov  a   »     =aPaih. 

162.  Similar  radical-surds  are  "  added  "  or  "  subtracted  " 
by  distributing  the  radical-surd  factor  with  the  coefficients. 
{Vide  §  73.)  If  possible,  first  reduce  by  the  principle  of 
Sections  159,  160,  e.g.,  1/3  V32  -  Vl8  +  3  -v'64  =  1/3 
V(16)  (2)  -  V(9H2)  +  3  V(4)  (2)  =  4  /3  V2  -  3  V2 
-\-Q-j2  =(4/3  -  3  -f  6)  V2  =  13/3  V2. 


106  NUMBER   AND   ITS   ALGEBRA. 

Staxements  involving  radicals  are  usually  intended  to 
concern  only  positive  roots  ;  but  in  abstract  operation  such 
statements  are  necessarily  various,  including  the  roots  in 
every  combination.  The  whole  truth  about  the  result  in 
the  example  is 

{±4/3  -  (±  3)  +  (±  6)}  V2  =  ±  13/3  V2,  or 
±5/3  V2,  or  -i-23/3  V2,  or  ±31/3  V2. 

The  V2  is  also  both  positive  and  negative;  but  since 
each  commensurable  factor  has  already  occurred  with  both 
signs,  no  new  value  would  be  obtained  from  the  double 
value  of  the  V2.  But  if  all  this  is  to  be  signified,  it  would 
be  better  to  be  explicit,  and  write  1/3  (zl=V32)  — (-j-VlS) 
+  3  (±-^64).     (Vide  §  120.) 

163.  Section  158  affords  the  rule  for  the  multiplication 
or  division  of  similar  radical-surds,  or  of  radical-surds  of 
the  same  order. 

If  radical-surds  are  not  of  the  same  order  they  may  be 
made  so  by  Section  160. 

The  Law  of  Indices  immediately  furnishes  the  rule  for 
the  involution  or  evolution  of  radical-surds. 

164.  The  student  should  exercise  himself  in  these  opera- 
tions. 

165.  Two  simple  binomial  quadratic  surds  are  called  con- 
jugate when  one  is  the  sum  and  the  other  the  difference  of 
the  same  two  terms  :.  e.g.,  a  -\-  -y/h  and  a  —  -y/h,  or  V«  + 
V^  and   -yja  —  -y/b. 

166.  Theorem.  —  The  product  of  conjugate  binomial 
quadratic  surds  is  a  stirpal  function  of  their  bases  (a  com- 
mensurate number  if  the  bases  are  commensurate  numbers), 
namely,  the  difference  of  the  squares  of  the  terras. 

Proof:  (Va  -f  V^*)  (Va  —  ^h)  =  a  +  V«  V^*  -  -\/h 
■yja  —  h  =  a  —  b. 


CLASSIFICATION   OF  ANALYTICAL   FUNCTION.      107 

167.  It  is  usuall}^  preferable  in  the  division  of  one  radi- 
cal-surd by  another,  or  of  a  commensurable  number  or 
non-radical  surd  by  a  radical-surd,  to  stirpalize  *  the  de- 
nominator. 

This  is  accomplished  when  the  divisor  is  a  monomial 
radical-surd,  as  Va"*,  by  multiplying  both  dividend  and 
divisor  by  -v/«"~"'.     For  example, — 

3      _     3  V2      _  ^  _c c  Va^      _ 

4V2  ~4V2V2~^/^^^'     bVa'~  hVa'Va^~ 

When  the  divisor  is  a  binomial  quadratic  surd,  multiply 
both  dividend  and  divisor  by  the  conjugate  quadratic  surd  ; 
when  a  trinomial,  make  it  a  binomial  by  association,  and 
apply  the  principle  twice. 

168.  Let  the  student  find  a  stirpalizing  multiplier  for 

This  is  the  most  general  case  of  a  monomial. 

169.  A  stirpal  integral  tervi  with  respect  to  any  num- 
bers, means  the  product  of  positive  integral  powers  of  those 
numbers. 

A  stirpal  integral  function  of  any  numbers  is  a  series 
(one  or  more)  of  stirpal  integral  terms  combined  in  addi- 
tion or  subtraction. 

Where  no  ambiguity  is  to  be  feared  we  may  say  merely 
"integral  function."  xj a  -\-  y fh  +  ;s/c  —  1  is  an  integral 
function  of  .r,  y,  z  ;  but  is  not  an  integral  function  of  a,  b,  c. 

In  integral  functions  the  degree  of  any  term  is  the  sum 

*  The  common  term  is  "  rationalize;  "  but  having  eschewed  this,  we 
must  say  stirpalize. 


108  NUMBER    AND   ITS   ALGEBRA. 

of  the  exponents  of  the  numbers  considered  (commonly 
called  variables)  ;  and  the  degree  of  the  function  is  the 
highest  of  the  degrees  of  its  terms. 

An  integral  function  of  the  1st  degree  is  often  called  a 
linear  function. 

The   term    degree    applies    only    to    integral    functions. 

Thus,  -  -I ^  +  1  is  of  no  degree  at  all :  the  term  does  not 

X        x^ 

apply. 

Functions  in  Avhich  the  variables  are  affected  by  positive, 
but  not  integral,  exponents  are  called  radical  functions. 
For  example,  a  +  -yjh  +  x,  or  a  +  (Z>  +  xf  ''\  is  a  radical 
function  of  x  (also  of  I)  ;  and  Vx  +  \j]),  or  {x  —  if  '^f '«, 
is  a  radical  function  of  x  and  y. 

Functions  in  which  the  variable  occurs  with  negative 
index  are  called  fractional  functions,  and  distinguished  as 
stirpal  or  radical  fractional  functions,  according  as  the  nega- 
tive index  is  integral  or  not.    Thus,  ^-,  or  «a;-^,  is  a  stirpal 

X 

fractional  function  of  a;;  and  ~,  or  ax-^'-,  is  a  radical 
fractional  function  of  x.  "^^ 

Integral,  radical,  and  fractional  functions  are  classed,  not 
very  felicitously,  as  "algebraical"  functions,  in  distinction 
from  others  equally  algebraical,  called  "  transcendental." 
I  shall  have  no  occasion  to  use  these  objectionable  terms, 
since  the  other  functions  are  all  particularly  named  upon 
their  own  merits. 

Functions  in  which  the  number  considered  occurs  as  an 
exponent  are  called  exponential  functions;  e.g.,  «^,  a-^"" 
are  exponential  functions  of  x. 

The  foregoing  classes  of  functions  are  those  organically 
involved  in  numerical  operations.     Others,  less  essentially 


FUNCTIONS   OF   EADICAL-SUEDS.  109 

connected  with  orgauic  laws,  are  named  from  their  several 
points  of  view ;  e.g.,  log  cc,  logarithmic  function ;  sin  a;, 
cos  X,  tan  x,  etc.,  trigonometric  functions,  etc. 

Numerical  functions  (Cf.  §§230,  234)  of  every  variety 
are  termed  analytical  functions  (Cf.  §§  145  and  156.) 

170.  Theorem.- — ^ Every  integral  function  of  quadratic 
surds  (V«,  V^,  Vc  .  .  .)  can  be  expressed  as  a  sum  of  a 
non-radical  term  and  multiples  or  fractions  of  the  radicals 
and  their  products  — 

(V'«5  V^,    V  c  .   .    .   's/iiO,   \ac,  -yhc  .   .   .   -y/abc  .   .   .). 

Proof:  Consider  any  integral  function  of  one  quad- 
ratic surd,  say  </>  (Va).  Terms  of  even  degree  are  non- 
radical, and  terms  of  odd  degree  can  all  be  reduced  to  the 
form  na"^  V«.  Collecting  the  even  and  odd  degree  terms, 
we  have  <^  ( Vc')  =  Z;  -)-  A  Va,  where  k  and  h  are  stirpal. 

If  we  have  <^  (V«,  V^),  proceeding  as  before,  we  get 
^  ( V«,  V^)  =  K  -\-  H  a/ a,  where  K  and  If  are  stirpal  so 
far  as  V^  is  concerned,  and  each. an  integral  function  of 
V^.  These  can  be  reduced,  and  will  yield  only  terms  such 
that      ^  (Va,  V^)  =  k  -{-  h  V«  +  '»i'  'Vb  +  n  ^/ab. 

171.  CoKOLLAKY.  — It  follows  that  <^  (—  V«)  ■=  k  —  h 
Va  ;  and  therefore  if  </>  Va  be  any  integral  function  of  Va, 
then,  </)  (  —  V«)  is  a  stirpalizing  factor  of  ^  V«.   {Cf.%  166.) 

Also  if  in  </>  (Va,  V^,  Vc,  .  .  .  )  we  change  the  sign  of 
any  one,  say,  V^,  then  ^  (Va,  V^,  Vc,  .  .  .  )  X  <^  (Va 
—  Vi,  Vc,  .  .  .  )  is  stirpal  so  far  as  V^  is  concerned. 

172.  Extension  of  the  theorem  to  all  stirpal  functions, 
integral  or  not,  of  quadratic  surds  —  and  of  the  corollary 

to  the  entire  stirpalization  of  ^  {-yja,  -\/b,  Vc,  .  .  .) is 

left  as  an  exercise  to  the  student. 

173.  As  a  very  simple  example  of  the  utility  of  these  prin- 
ciples, suppose  one  had  to  calculate  to  five  decimal  places, 


110  NUMBER   AND    ITS   ALGEBRA. 

1 .     Time  and  labor  would  be  saved  by  redu- 

1  -)-  V2  -|-  V3  cing  to  the  equivalent  integral  function 
of  the  radicals,  1/2  +  1/4  V2  -  1/4  V6,  before  cal- 
culating. 

174.  Theorem.  —  '^  If  p,  q,  A,  B,  be  all  commensurable, 
and  V/?  and  Vg"  incommensurable,  then  we  cannot  have 
Vi>  =  A  +  B  Vq. 

"  For,  squaring,  we  should  have,  as  a  consequence,  j)  =  A^ 

-\-  B"^  q  -\-  2  AB  V«7 ;  whence,  Vv  =  —^ ~ ,  which 

-r        i  -r  I,  ,      1  2AB 

asserts,  contrary  to  our  hypothesis,  that  -y/q  is  commen- 
surable." 

The  proof  of  this  theorem,  Avhich  is  copied  verbatim 
from  Chrystal's  Text  Book  of  Ahjebra,  Vol.  I,  p.  200,  estab- 
lishes what  may  seem  at  first  sight  a  contradiction  of  the 
doctrine  of  the  Continuity  of  Number.  Especially  so,  under 
the  somewhat  ambiguous  title  of  the  section  in  Professor 
Chrystal's  work  (perhaps  the  best  yet  written  in  English), 
the  ^^Independence  of  Surd  Numhers.^^  Eadical-surds  are 
definite  parts  of  the  continuous  magnitude,  Number ;  nor 
does  the  theorem  contradict  this ;  nor  are  radical-surds 
<'  independent "  in  any  other  sense  than  that  there  are  no 
commensurable  numbers  such  that  V^^  =  A  -{-  B  ^  q. 

175.  Since,  by  Section  170,  any  integral  function  of  a 
quadratic  surd  can  be  expressed  as  in  the  form,  A  -\-  B  -y/q, 
it  follows  from  Section  174  that  one  quadratic  surd  cannot 
be  expressed  as  an  integral  function  of  a  dissimilar  surd. 

176.  It  is  an  obvious  corollary  of  Section  174  that  if  A;  + 
h  -yj a  -\-  m  V^  +  n  "y/ab  =  0,  where  neither  a  nor  b  is  zero, 
then  k  =  0,  A  =  0,  m  =  0,  and  n  =  0. 

177.  One  case,  whose  utility  is  experienced  very  early 
in  algebraic  studies  deserves  special  mention.      If  a  -\-  Va; 


NEOMONTC    NUMBER.  Ill 

=  b  -\-  -y/i/,  then  a  =  h  and  x  =  y,  provided  a,  b,  x,  and  y 
are  all  commensurable,  and  -\/x  and  V*/  surds. 

178.  Let  the  student  prove  that  the  product  or  quotient 
of  two  similar  quadratic  surds  is  commensurable ;  and 
inversely. 

The  like  is  not  true  for  radical-surds  of  higher  orders  ; 
but  let  him  show  that  the  product  of  two  similar,  or  of  two 
equiradical,  surds  is  either  commensurable  or  an  equirad- 
ical  surd. 

179.  We  are  now  prepared  to  take  up  the  consideration 
of  the  problem  presented  in  a}''^  where  a  is  negative,  and 
n  an  even,  positive  integer. 

As  Ave  saw  in  Section  154,  the  operation  is  unintelligi- 
ble under  the  concept  of  Number  thus  far  attained.  But 
if  the  Principle  of  Continuity  is  valid,  the  result  must  be 
a  number ;  and  if  not  any  number  hitherto  conceived,  we 
must  investigate  the  characteristics  of  this  unknown  num- 
ber, X  in  the  synthetic  equation  (—  1)  Y^  =  x. 

180.  Whether  fortunately  or  unfortunately,  this  prob- 
lem confronts  pupil  and  teacher  at  a  very  elementary 
stage  of  numerical  analysis.  In  every  high  school  the 
solution  of  quadratic  equations  is  attempted;  and  these, 
even  in  the  simplest  form,  are  in  general  solvable  only  in 
terms  of  neomonic  and  complex  numbers.  The  question, 
therefore,  cannot  be  postponed ;  and  it  behooves  every 
teacher  to  clear  up  his  ideas  on  this  subject. 

181.  Mathematicians  of  to-day  have  left  the  point  of 
view  of  the  sixteenth  century,  from  which  numbers  were 
characterized  as  "  rational  "  and  "  irrational,"  "  real  "  and 
"imaginary  ;  "  they  use  V—  1  as  naturally  as  —  1.  Neo- 
monic one,  and  negative  one,  bear  a  similar  relation  to 
Primary  Number. 


112  NUMBER   AND   ITS   ALGEBRA. 

The  conception  of  neomonic  number  is  not  essentially 
more  difficult  than  that  of  negative  number.  He  who  can 
conceive  the  one,  can  conceive  the  other.  The  V— 1  is  no 
more  an  impossible  and  meaningless  operation  in  terms  of 
protomonic  number,  than  1  —  2  is  impossible  and  unintel- 
ligible in  terms  of  primary  number.  Terms  are  often  bab- 
bled in  unconscious  vacuity  of  thought.  Many  speak  quite 
familiarly  of  negative  number,  who  nevertheless  regard 
neomonic  number  as  some  irrational  and  meaningless  trick 
of  handwriting.  As  suggested  in  Chapter  XII,  I  lament 
imperfect  concejits  of  Number  on  the  imrt  of  us  all,  but  let 
no  man  pigeon-hole  in  his  mind  contradictory  opinions.  It 
seems  to  me  something  to  put  neomonic  numbers  on  the 
same  footing  as  negative  numbers,  or  even  numerical  frac- 
tions. 

When  this  point  of  view  is  attained,  I  think  we  stand 
in  the  dawn ;  or  rather  that  the  sun  has  risen  upon  Arith- 
metic, even  as  it  has  risen  upon  Geometry.  Perhaps  we 
shall  not  have  long  to  wait  for  still  fuller  and  more  satis- 
fying interpretations  of  number  than  have  been  expounded 
hitherto ;  because  not  one  man,  but  hundreds,  have  reached 
some  such  standpoint  as  that  from  which  I  have  endeav- 
ored to  present  the  subject.  During  two  thousand  years 
after  Euclid  saw  that  he  must  assume  the  "  parallel  postu- 
late "  it  was  universally  regarded  either  as  an  axiom,  or  as 
a  theorem  capable  of  demonstration.  But  finally  the  true 
insight  was  gained  (regained)  by  many  minds  about  the 
same  time ;  and  then  the  Non-Euclidean  Geometry,  and 
daylight  became,  indeed,  "inevitable."* 

*  The  MoJiist,  July,  1894,  ' Xon-Eudidean  Geometry  Inevitable,  by- 
George  Bruce  Halsted.  Of  course  tVie  majority  of  text-books  still  pre- 
sent Geometry  at  this  crucial  place  from  the  mediaoval  standpoint  ;  but 


THE   NEOMON.  113 

182.    If  the  V  —  1  is  a  number,  we  have  by  definition 

V-1  V-l=  -1, 
also  Va  ^a  =  a,  where  a  is  any  positive  protomonic 

number ; 
therefore     (Va  V—  1)   (V«  V—  1)  =  —  a,    multiplying 

member  by  member ; 
therefore  Va  V—  1  =  V—  «,  taking  square  root  of  each 

member. 

Consequently  it  appears  that  the  square  root  of  any 
negative  number  is  the  product  of  the  square  root  of  the 
corresponding  positive  number  and  V  —  1-  Considering 
also  all  multiples  and  fractions  of  V—  1,  and  the  nega- 
tives of  each,  we  discern  a  continuous  Number  whose  unit 
is  V—  1>  and  which  has,  therefore,  been  called  Neomonic 
Number.  The  Number  whose  unit  is  1  may  be  called  Pro- 
tomonic in  contradistinction. 

Writing  i  for  V—  Ij  this  continuous  series  may  be  rep- 
resented — 

—  cx>  i  .    .    —  2  i  .   .    —  -y/2  i  .  .  —  i  .   .    —  ^  i  .   .   0  (i)  .   , 
-f  ^  i  .  .  .  -\-  i  .  .   +  V2  i  .  .   +  2  t  .  .   +  CO  i. 

The  protomonic  series  may  be  represented  — 


—  CO 


.   -  2  .  .  .   -  V2  .  .  .  -  1 .  .   -  1/2 .  .  0     ... 

-f  1/2  .  .   +  1  .  .   +  V2  .  .  .   +  2  .  .  .   +  00 . 

183.    No  neomonic  number  can  equal    any  protomonic 
number  except  0  i  =  0.     For  it  is  deducible  from  various 

this  is  probably  as  much  due  to  the  mercantile  rule  of  using  up  a  stock- 
on-liand  before  advancing  to  something  better,  as  to  ignorance  of  recent 
developments.  No  doubt  hundreds  of  teachers  put  the  "  axiom  "  in  its 
right  place  in  their  expositions  of  the  text ;  and  so,  as  it  were  by  a  note, 
bring  their  text-books  "  up  to  date." 


114  NUMBER   AND   ITS   ALGEBRA. 

premises  that  0  1  =  0.  Thus,  ii  xl  =  0,  then  (xi)  (xi)  =  0, 
that  is,  —  x^  =  0;  therefore  x  =  0,  and  therefore  0  *  =  0. 

184.    Most  laws  of    operation  with   neomonic   numbers 
are  evident  from  familiar  princixDles.     Thus:  — 

ai  4-  bi  =  (a  -\-  h)  i  .  .  .  hence  the  sum  of  two  neomonic 
numbers  is  neomonic. 

ai  —  hi  =  (a  —  b)  i  .  .  .  hence  the  difference  of  two  neo- 
monic numbers  is  neomonic. 

ai  X  b  =  abl  .  .  .  hence  the  product  of  a  neomonic  and  a 
protomonic  number  is  neomonic. 

ai  X  bi  =  —  ab  .  .  .  hence  the  product  of  two  neomonic 
numbers  is  protomonic. 

ai  -ir  b  r=  (a  1 0)1  i  _  ,  hence  ratios  of  protomonic  and 
h  -r-  ai  =  (^—  b  I  a)  I  )  neomonic  numbers  are  neomonic. 
ai  ^  bl  =  a  I  b  .  .  .  hence  ratios  of  neomonic  numbers  are 

protomonic. 
^2  =  -  1 ;  P  =  -  1  V-  1  =  -  * ;   ^^  =  *2  r  _  +  1  ;    and 

where  n  is  a  positive  integer, 

(aiy  =  {ai  .  ai  .  .  .  n  factors)  =  {aaa  .  .  .  n  factors) 
(iii  .  .  .  n  factors)  =  a"*", 

that  is,  the  positive  integral  power  of  a  neomonic  number 
is  protomonic  or  neomonic  according  as  the  same  power  of 
i  is  protomonic  or  neomonic.  Moreover,  the  integral  powers 
of  *  are  seen  to  recur  in  a  period  or  cycle  of  four  different 
values.  Negative  exponents  result  as  always  in  the  recip- 
rocal of  the  same  number  Avith  like  positive  exponent. 

185.    Discussion  of  radical  functions  of  i,  and  the  inter- 
pretation  of   neomonic  exponents,  is  postponed  to  more 


COMPLEX   NUMBER.  115 

advanced  studies  ;  but  we  are  not  led  to  any  new  applica- 
tion of  the  principle  of  Continuity,  and  therefore  to  no  new 
mode  of  jSTumber,  beyond  the  result  of  combining  proto- 
monic  and  neomonic  numbers  in  addition  and  subtraction. 

186.  The  extension  of  the  number-concept  reaches  its 
own  essential  terminus  in  the  operation  a  +  hi,  where  a 
and  h  are  protomonic. 

In  a -{-hi  we  have  the  most  general  expression  of  num- 
ber ;  for  it  is  protomonic,  neomonic,  or  complex,  according 
as  ^  =  0,  a  =  0,  or  neither  equals  0. 

187.  The  result  of  the  operation  a  -\-  hi,  is  called  a  com- 
plex number ;  and  is  seen  to  be  really  a  new  mode  of  Num- 
ber by  considering  the  series  of  complex  numbers  formed 
in  a  +  hi,  as  a  and  h  pass  independently  through  all  pro- 
tomonic values. 

188.  It  is  highly  important  to  note  this  two-fold,  two- 
dimensional  (vide  §  229,  et  seq.),  character  of  complex 
number,  and  its  consequent  contrast  with  protomonic  and 
neomonic  number.  There  is  only  one  way  of  varying  x 
continuously  (without  repetition  of  intermediate  values) 
from  —  2  to  +  3,  if  it  remains  protomonic.  Likewise,  only 
one  way  for  continuous  passage  of  x  from  —  2  i  to  -\-  3  i, 
if  it  is  to  be  always  neomonic.  But  in  utter  contrast,  there 
is  an  infinite  variety  of  ways  for  x  to  pass  continuously 
from  —  2  +  3  i  to  +  2  +  3  /,  remaining  always  a  complex 
number.      (Vide  §  197.) 

189.  If  a  =  0  and  b  =  0,  a  -\-  hi  =  0;  and  inversely. 

190.  Complex  number  contains  all  protomonic  and  all 
neomonic  number  as  special  cases,  and  is  therefore  Number 
in  its  final  generalization. 

191.  The  student  should  everywhere  carefully  avoid  con- 
fusion in  dealing  with  the  alternate  square  roots  of  any 


116  NUJIBEE,   AND   ITS   ALGEBRA. 

number ;  but  especially  is  this  the  case  with  neoinonic 
numbers.  Having  been  accustomed  to  write  {vide  §§  64, 
158)  Va  V6  =  V"^,  he  may  fall  into  the  error  of  writing 
V—  a  V^^  =  V(—  «)  (—  h)  =  -y/ah.  I  call  this  an  error 
because  we  must  be  consistent  in  algebraic  conventions  ;  and 
in  such  contexts  the  positive  root  is  understood  by  -y/ab* 

It  is  not  a  true  statement  that  Vf'-  V^  =  Vo^,  if  the 
square  roots  are  to  be  taken  at  random.  One  cannot  make 
various  assertions  in  the  same  sentence.  Therefore,  in 
Va  V^  =  Vab,  we  evidently  mean  only  the  positive 
square  roots  to  be  considered.  If  negative  roots  are  to  be 
taken  into  account,  we  must  say  what  we  mean.  Thus 
(writing  -j-  V*  for  j^ositive  square  root  of  a,  and  —  V«  for 
negative  square  root  of  a)  (—  V«)  (—  V^)  =  +  'Vab;  or 
(—  V")  (+  "v^)  =  —  Vcib,  etc. 

Now,  if  in  accordance  with  the  algebraic  convention 
plainly  exhibited  above,  we  consider  only  positive  square 

*  In  a  translation  just  published  of  Durege's  Theory  of  Functions  of 
a  Complex  Variable,  by  Professors  Fischer  and  Schwatt  of  the  Univer- 
sity of  Pennsylvania,  Philadelpliia,  189(3,  it  is  stated  on  Page  10  of  the 
Introduction:  "Euler  himself  taught,  as  now  generally  accepted,  that, 
if  a  and  b  denote  two  positive  quantities,  V—  «  V^^  =  y/ab ;  i.e.,  that 
the  product  of  two  imaginary  quantities  is  equal  to  a  real  quantity." 

The  omission  of  the  minus  sign  before  Vab  may  be  a  typographical 
error;  for  the  authors,  like  all  others,  use  \/— a  ■\/^^  =  —  Vab. 

In  the  translators'  Introduction  it  is  very  appropriately  remarked:  — 

"To  follow  the  gradual  development  of  the  theory  of  imaginary  quantities 
is  especially  interesting,  for  the  reason  that  we  clearly  perceive  with  what  diffi- 
culties is  attended  the  introduction  of  ideas,  either  not  at  aU  known  before,  or 
at  least  not  sufficiently  current.  The  times  at  Avhich  negative,  fractional,  and 
irrational  quantities  were  introduce.l  into  mathematics  are  so  far  removed  from 
ns,  that  we  can  form  no  adequate  conception  of  the  difficulties  which  the  intro- 
duction of  those  quantities  may  have  encountered.  Moreover,  the  knowledge 
of  the  nature  of  imagi)iiiry  quantities  has  helped  us  to  a  better  understanding 
of  negative,  fractional,  and  irrational  quantities,  a  common  bond  closely  unit- 
ing them  all." 

Of  course  I  would  have  one  read  numbers  in  the  place  of  "  quantities." 


EFFICACY   OF   NUMERICAL   OPERATIONS.  117 


roots  of  neomonic  numbers,  V—  «  V—  ^  does  not  equal 
■\/ab,  but  —  V«6 ;  — 

for  V—  a  V—  b  =^ai^bi  =i^^ab  =(  — 1) V«^  =  —  Vf<6. 
One  need  find  no  difficulty  in  reconciling  with  the  Prin- 
ciple of  Continuity  the  statements  that,  regarding  only 
positive  roots,  Va  V^  =  -s/ab,  while  V—  «  V—  *  is  not 
equal  to  V(—  «)  (—  b).  The  law  of  indices  must  be 
applied  with  due  regard  to  other  laws.  The  essential 
statement  of  the  law  of  indices  is  «^  av  =  a^  +  y.  This 
includes  all  particular  cases  as  a,  x,  and  y  assume  differ- 
ent characters.  But  it  has  been  necessary  with  every 
phase  of  number  to  understand  in  this  statement  that  only 
corresponding  roots  are  considered  when  x  and  y  are  frac- 
tional with  even  denominators.  (Cy.  §§  144,  146.)  For 
example,  V'^  X  1^^^  would  not  equal  11/2  +  1/2^  ^^  \^  if 
one  positive  and  one  negative  root  were  taken.  Now, 
this   fundamental   statement   of  the   law  of  indices  hokls 

for  all  number.     It  is  the  very  definition  of  V—  1,  that 
(_  1)W2  (^_  iy/2  ^  ^_  ;Ly/2  +  i/'2^  (-_  ly  =  _  1. 

It  was  easily  proved  for  protomonic  number  that,  regard- 
ing only  corresponding  roots  when  a;  is  a  fraction  with 
even  denominator,  a^  a^  =  («ff)^,  and  a'-'  b^  =  (abY ;  but 
when  a  and  aa  differ  in  quality,  the  very  conditions  of 
the  original  statement  are  abolished  (it  is  as  if  one  posi- 
tive and  one  negative  root  of  a^  had  been  taken),  and 
different  conclusions  might  be  anticipated  under  the  same 
laws. 

In  fine,  all  this  is  not  an  anomaly  of  V—  1  in  operation, 
but  merely  an  alternative  statement  of  its  existence.  The 
difficulty  lies  in  the  origin  of  neomonic  number,  not  in  its 
operation. 

On  the  other  hand,  a^/b'^  =  (ct  /  by,  established  for  pro- 


118  NUMBER   AND   ITS   ALGEBEA. 

tomonic  number,  does  hold  if  a^  and  h'^  are  neomonic,  — ■ 
simply  because,  in  this  case,  no  qualitative  difference  arises 
in  the  direct  performance  of  the  operations  indicated  by 
the  two  members  of  the  equation,  if,  in  accordance  with 
the  meaning  of  the  formula,  only  positive  roots  are 
regarded. 

For  example,  (-  ^Y'-i  (-  9)^  ^2=  (4/9)^ '^ ;  for  (-  4)^  '^ 
=  2  i,  and  (-  9)7^  =  3  *  ;  therefore,  (-  4)77  (_  9)^  '''- 
=  2ilZi  =  2j2>.  Also  the  positive  square  root  of  4/9 
is  2  /  3.  

Note,  also,  that  for  a  like  reason  V«  V—  h  =  V—  ab; 
for  -y/a  -y/ —  6  =  -y/a  -\b  i  =  '\ab  i,  and  V —  cih  =  ~vab  i. 

The  safe  practice  is  to  express  every  neomonic  number 
in  its  essentially  proper  form,  as  based  upon  a  new  unit. 

Rules  of  thumb  would  conduct  one  to  true  results  in  all 
operations  except  multiplication ;  but  for  many  reasons, 
always  express  -y/—  a  as  -\/at.  If  you  do  this,  correct 
calculation  will  be  easy  under  the  very  definition  of  the 
neomon,  i^  =  —  1. 

192.  As  a  natural  consequence  of  the  view  that  Algebra 
is  some  mysterious  conglomeration  of  "  pure  symbols " 
(C/l  Introduction,  pp.  8,  12)  without  content,  existing  for 
itself,  void  of  numerical  meaning,  it  was  long  discussed, 
as  if  it  were  a  matter  to  be  settled  by  parliament,  whether 
V—  (f^  V —  b  should  equal  V—  «^,  or  —  V«^.  Only  one 
hundred  years  ago  English  mathematicians  were  divided 
on  this  question.  One  party  argued  that  the  product  must 
be  V —  ab;  because  the  product  of  one  "impossible  quan- 
tity "  by  another,  could  not  possibly  equal  a  "  real 
quantity "  —  as  if  a  priori  deduction  of  Avhat  is,  or  is 
not,  possible  with  imjjossihle  quantities  was  not  ab  initio 
an  impossible  discussion  within  the  realm  of  Reason. 


COMPLEX  NUMBERS.  119 

May  not  tlie  foregoing  discussion  (as  well  as  every  other 
investigation  we  have  pursued)  serve  to  emphasize  the  car- 
dinal thesis  of  these  lectures ;  namely,  that  the  essential 
nature  of  any  algebra  is  as  defined  in  Section  20 ;  that  it  is 
Arithmetic,  as  the  science  of  Number,  which  everywhere 
underlies,  shapes,  and  organizes  our  Algebra ;  that  it  is 
real  numerical  laws  and  operations  that  the  algebra  conven- 
tionally expresses;  that,  although  Number  is  certainly  a 
creation  of  the  human  intellect,  it  is  not,  therefore,  the 
creature  of  our  choice  or  whim;  that,  once  formed,  the 
Idea  unfolds  itself ;  that  every  numerical  problem  is  a 
question  of  Truth ;  that  the  explanation  is  to  be  discov- 
ered; and  that  the  verdict  is  nowise  subject  to  conven- 
tional decision  or  parliamentary  settlement. 

193.  It  might  be  very  helpful  to  illustrate  the  proper- 
ties of  complex  number  by  the  graphic  representation 
known  as  Argand's  diagram,  which  constitutes  the  foun- 
dation of  a  beautiful  application  to  geometry;  but  we 
shall  here  confine  ourselves  to  purely  analytical  inves- 
tigations. 

We  have  seen  (§  188)  the  two-dimensional  nature  of 
complex  number,  and  the  infinite  variety  of  ways  in 
which  it  may  vary  continuously  from  a  -\-  hi  to  c  -f  di, 
because  the  protomonic  and  neomonic  parts  may  vary 
independently. 

In  order  that  x  -\-  yl  shall  become  zero,  x  and  y  must 
vanish  simultaneously.  Por,  li  x  -\-  yi  =  0,  ic  =  —  yi,  and 
hence  cc  =  0  and  y  =  0,  —  else  would  a  protomonic  number 
equal  a  neomonic,  Avhich  is  impossible  except  both  be  zero. 
(Vide  §  183.) 

On  the  other  hand,  if  either  x  or  y  becomes  infinite,  x  -{- 
yi=cc.     (FicZe§198). 


120  NUMBER   AND   ITS   ALGEBRA. 

194.  li  a  -\-  bi  =  G  -\-  di,  then  a  =  c  and  b  =  d. 

For,  subtracting  c  -f-  di  from  each  member  of  the  given 
equation,  a  —  c-\-(b  —  d)i  =  0;  therefore,  by  Section  193, 
d  —  c  =  0  and  b  —  d  ^=  0  ;  that  is,  a  =  c  and  b  =  d. 

Of  course,  since  x  +  (—  y)  =  x  —  (+  y)  ox  x  —  y  {%  120), 
the  preceding  formula  inckides  all  combinations  as  a,  b,  c, 
and  d  are  positive  or  negative ;  e.g.,  if  a  -f  bi  =  c  —  di, 
a  =  c,  b  =  —  d. 

195.  Two  complex  numbers  which  differ  only  in  that 
one  is  the  result  of  the  addition,  the  other  of  subtraction, 
of  the  neomonic  part,  are  called  conjugate  ;  e.g.,  —  1  /  2  +  2  i 
and  —  1/2  —  2i,  0T2i  and  —  2 i,  or  generally  x  -\-  yi  and 
X  —  yi. 

Obviously  the  sum  of  conjugate  complex  numbers  is  pro- 
tomonic,  but  so  also  is  their  product :  — 

(x  -j-  yi)  (x  —  yi)  =  x^  —  y"^  i"^  =  x"^  -[-  y^. 

196.  Let  the  student  prove  the  inverse  proposition. 

197.  x^  4-  ?/^  is  called  the  norm  of  the  complex  number 
X  -|-  yi,  ox  X  —  yi]  and,  as  seen  in  Section  195,  the  product 
of  conjugate  complex  numbers  is  the  norm  of  each. 

But  note  that  also  norm  (—  a?  —  yi)  =  (—  xy  -\-  y'^  = 
x^  +  if'j  although  —  X  —  yi  is  not  conjugate  with  x  +  yi, 
nor  is  their  product  the  norm  of  either;  for  (—  x  —yi) 
(x  -\-  yi)  =  y"^  —  x"^  —  2  xyi. 

198.  The  positive  square  root  of  the  norm  of  a  complex 
number  is  called  its  modulus  :  mod  (x  +  yi)  =  +  Vic^  +  y'^. 

This  modulus  has  extremely  important  properties. 

The  attentive  student  may  have  already  discerned  diffi- 
culty in  applying  comparisons  of  greater  or  less  to  complex 
numbers ;  for  example,  which  is  the  greater,  3  +  4  i  or 
2  +  5i? 


MODULUS   OF  COMPLEX  NUMBER.  121 

The  quantity  {vide  §  229)  of  a  complex  number  is  discov- 
ered to  depend  upon  its  modulus.  Complex  numbers  with 
equal  moduli  are  quantitatively  equal,  though  not  identical 
numbers.  Any  magnitude  of  two  dimensions  must  exhibit 
this  mode  of  equivalence  without  congruence.  Argand's 
diagram  would  give  a  good  illustration  of  this  relation  : 
the  points  representing  (or  terminating  the  radii  which 
represent)  complex  numbers  of  equal  moduli  would  all  lie 
on  a  circle  ;  points  corresponding  to  complex  numbers  of 
less  moduli  would  lie  within  the  circle,  and  of  greater 
moduli  without. 

This  property  of  the  modulus  is  exhibited  analytically  in 
the  fact  that,  since  mod  {x  +  yi)  =  +  V^^  +  y^  which  is 
positive  regardless  of  the  quality  of  x  or  y,  if  either  x  ox  y 
increases,  the  modulus  increases,  and  if  either  x  ox  y  de- 
creases, the  modulus  decreases.  And  this  change  is  continu- 
ous, the  modulus  vanishing  with  the  number,  and  inversely. 

If  two  numbers  are  equal,  their  moduli  are  equal ;  for 
we  have  seen  (§  194),  \i  a  -\- hi  =  c  -\-  di,  a  =  c  and  b  =  d. 
But  the  inverse  is  not  true  ;  for  if  cv'  -{-  b'^  r=  d^  -\-  d%  it 
does  not  follow  that  a  =  c  and  b  =  d.  ^ 

Note  that  if  7/  =  0  in  x  +  yi,  that  is,  if  the  complex 
number  be  wholly  protomonic,  the  modulus  becomes  +  Vx^ 
=  _|_  X,  —  and  this  whether  x  in  the  complex  number  be 
positive  or  negative.  Thus,  the  mod  (+  3)  =  +  V(+  3)"'^ 
=  +  3  ;  and  mod  (-  3)  =  +  V(-  3)^  =  +  3. 

For  this  reason,  many  European  continental  writers  use 
the  term  modulus  of  x  ('^mod  x")  where  a:;  is  a  protomonic 
number,  instead  of  the  term  "  numerical  value  of  a;,"  em- 
ployed by  English  writers.  For  example,  we  constantly 
speak  of  +  3  and  —  3  as  "numerically  equal,"  whereas, 
if  equal  —  being  numbers  —  they  could  only  be  numerically 


122  NUMBER   AND   ITS   ALGEBEA. 

equal ;  and  they  are  not  equal,  for  their  difference,  instead 
of  being  zero,  is  6. 

It  would  therefore  serve  accuracy  and  propriety  to  fol- 
low the  practice  of  the  writers  referred  to. 

199.  Evidently  the  sum  of  any  number  of  complex  num- 
bers is  a  complex  number. 

Likewise  the  product  of  any  number  of  complex  num- 
bers is  a  complex  number. 

Also  the  ratio  of  two  complex  numbers  is  a  complex 
number.     For  — 

a  -\-  bi  _  (a  -\-  hi)  (c  —  di)  _  (ac  -\-  hd)  —  (ad  —  cb)  i 
c  +  di  ~  c2  +  cZ-  ~  ~c2  +  d^ 

_  fac-\-  bd\  _ 

-\c^^d^' 

which  is  a  complex  number. 

Since  stirpal  functions  can  involve  only  the  operations 
of  addition,  multiplication,  and  their  inverses,  we  have 
thus  established  the  theorem  :  Every  stirpal  function  of 
one  or  more  complex  numbers  is  a  complex  number. 

200.  Several  other  fundamental  theorems  concerning 
stirpal  functions  of  complex  numbers,  and  moduli  of  com- 
plex numbers,  are  deferred  to  the  final  chapter. 

In  regard  to  radical  functions  of  complex  numbers,  we 
can  consider  here  only  the  particular  case  of  the  square 
root :  — 

Assume  that  the  square  root  of  a  complex  number  is  a 
complex  number,  and  let  — 

V«  +  bi  =  A  +  Bi, 

where  a,  b,  A,  and  B  are  protomonic. 

Squaring  each  member  :  a  -\-  bi  =  A"^  —  B^  -\-  2  ABi ; 


EFFICACY   OF   NUMERICAL   OPERATIONS.  123 

therefore,  by  Section  194,    a  =  A^  —  B^  (1) 

and  b  =  2AB.  (2) 

Adding  the  squares  of  (1)  and  (2) 

a2  -|_  J2  =  (^2  _^  ^2y^  (3>) 

Taking    square    roots    of    (3),    and    remembering    that 
j2  _|_  ^2  jg  essentially  positive  : 

Adding  (1)  and  (4)  :     +  V«M^  -\-a  =  2A^ 


therefore 


=  .v/ 


+  Va"-^  +  b'^  + 


a 


2 
Subtracting  (1)  from  (4)  :  +  Va^  +  b'^  —  a  =  2  B% 

therefore  ^  =  ±  y/+  Va2  +  6^  _^ 


Since  -J-  Va'^  +  ^^  >  «>  these  values  of  A  and  i>  are  pro- 
tomonic.  Since  b  =  2  AB,  like  signs  in  the  values  of  A 
and  B  must  be  taken  if  b  is  positive,  and  unlike,  if  b  is 
negative,  therefore  if  b  is  positive, 


v;r+l5=±{v/+^ffl±i!+.-y/+Vi!±^!^|     r. 

and  if  b  is  negative, 

Let  the  student  verify  by  multiplication. 

If  the  protomonic  part  of  the  complex  number  vanish, 
we  have  here  the  formula  for  the  square  root  of  a  neomonic 
number. 

Particularly  if  a  =  0,  and  b  =  -\-l,  formula  I  becomes,  — 
/ -.  1  +  i 


124  NUMBER   AND   ITS   ALGEBRA. 

If  a  =  0,  and  &  =  —  1,  we  have  from  formula  II,  — 

1  -i 


V-^=  ± 


V2 


By  means  of  these  results  the  student  can  readily  find 
four  4th  roots  of  -|-  1>  and  of  —  1. 

201.  Since  radical  functions  of  any  number  involve  only 
fractional  exponents  besides  stirpal  operations  upon  the 
number,  if  we  show  that  the  nth.  roots,  when  w  is  a  primary 
number,  of  a  complex  number  are  complex  numbers,  we 
establish  the  theorem :  All  radical  functions  of  complex 
numbers  are  complex  numbers. 

The  investigation  must  be  postponed  to  future  studies  ; 
for  more  powerful  instruments  of  analysis  (e.g.,  Demoivre's 
theorem)  are  required  than  are  at  the  command  of  the 
students  to  whom  these  lectures  are  primarily  addressed. 
But  the  theorem  has  been  established. 

202.  Command  of  the  proper  means  of  analysis  (e.g., 
logarithmic  series)  would  enable  the  student  to  prove  that 
exponential  functions  (^vide  §  169)  of  complex  numbers 
lead  to  no  new  mode  of  number. 

Thus,  finally,  it  has  appeared  that  the  ultimate  gener- 
alizatioii,  (a  -\-  hiY+y\  is  still  a  complex  number;  and  that 
therefore  the  Universe  of  Number  closes,  returns  upon 
itself,  is  comjjlete. 


MEASUREMENT.  125 

XITI.     Measurement. 

203.  The  measurement  of  any  magnitude  (concrete  or 
abstract)  is  the  process  of  finding  its  ratio  to  another  mag- 
nitude of  the  same  kind,  arbitrarily  chosen  as  a  unit. 

204.  The  measure  of  a  magnitude  is  this  ratio  —  a 
number. 

Under  the  conventions  of  English  speech,  the  measure 
of  any  magnitude  is  expressed  by  a  phrase  made  up  of  this 
number  and  the  name  of  the  chosen  unit. 

205.  The  noun,  measure,  is  commonly  used  in  the  sense 
of  Section  204,  in  the  sense  of  submultiple  {vide  §  83),  and 
in  the  sense  of  unit.  There  may  be  no  very  good  ground 
of  choice  in  these  terms,  but  consistency  in  the  same  dis- 
course is  desirable.  It  may  be  better  not  to  say,  '*  the 
greatest  common  measure "  of  two  or  more  magnitudes, 
since  any  magnitude  of  the  same  kind  woiild  be  a  common 
measure,  in  another  meaning  of  the  word  ;  for  example, 
the  yard  may  be  the  common  measure  of  all  lines,  and  so 
may  the  metre.  It  may  be  better  to  say,  instead,  the 
greatest  common  submultiple.  On  the  other  hand,  commen- 
surable and  incommensurable  point  the  same  way  as  common 
measure. 

The  use  of  measure  in  the  sense  of  unit  is  superfluous 
in  the  presence  of  the  clearer  term,  unit,  and  appears  to 
foster  a  confusion  of  concepts  with  commensurability ; 
whereas  it  is  very  seldom  that  a  unit  is  commensurable 
with  the  magnitude  measured.  Attention  is  merely  called 
to  this  confusion  in  our  language,  and  consequently  in  our 
thought.  Under  the  necessity  of  some  choice,  I  have  se- 
lected cornvfiensurahle,  submultiple.,  and  unit,  and  have  simply 
avoided  measure  as  the  inconsistent  synonym  both  of  sub- 
multiple  and  unit. 


126  NUMBER   AND  ITS   ALGEBRA. 

206.  The  unit  of  any  kind  of  magnitude  may  be  any 
magnitude  of  the  same  kind. 

Convenience,  or  lack  of  concerted  choice,  ofteia  establishes 
in  common  usage  many  units  for  magnitudes  of  the  same 
kind. 

207.  Magnitudes  are  of  the  same  kind  when,  of  any  two, 
one  is  necessarily  greater  than,  equal  to,  or  less  than  the 
other. 

Magnitudes  between  which  there  is  no  such  comparison 
are  of  different  kinds ;  and  between  such  there  is  no  ratio, 
nor  could  one  be  added  to  the  other. 

208.  The  ratio  of  any  two  magnitudes  is  independent  of 
any  unit,  or  units,  of  measurement.  Their  absolute  values 
can  in  no  way  depend  upon  the  arbitrary  standard,  or  stan- 
dards, by  which  they  may  happen  to  be  estimated.  For 
example,  the  ratio  of  the  time  of  rotation  of  Mars  to  the 
time  of  rotation  of  Venus  is  that  exact  numerical  relation 
of  the  former  to  the  latter,  in  virtue  of  which  the  former  is 
a  fraction  of  the  latter ;  or  greater  than  one,  and  less  than 
another  fraction  of  the  latter,  which  differ  as  little  as  we 
please.  (Cf.^  83.)  Evidently  this  ratio  can  nowise  de- 
pend upon  other  comparisons  of  these  times  with  any  other 
periods  of  time  whatsoever. 

209.  But  the  ratio  of  any  two  magnitudes  equals  the 
ratio  of  their  respective  measures  in  comparison  with  the 
same  unit.  For  example,  the  ratio  of  the  two  periods  of 
planetary  rotation  just  mentioned  equals  the  ratio  of  their 
respective  ratios  to  any  third  period  of  time,  say,  the  time 
of  the  earth's  rotation,  ■ —  tlie  period  we  name  a  day. 

210.  There  is  such  a  thing  as  direct  operation  with  con- 
crete magnitudes  ;  but  it  is  only  through  their  measures, 
that  is,  their  ratios  to  some  unit,  that  magnitudes  other 


MEASUREMENT    BY   PROPORTIONALITY.  127 

than  Number  can  become  subjects  of  genuine  calculation, 
the  proper  subjects  of  which  are  numbers,  and  numbers 
alone.  {Cf.  §§  27  and  48.)  For  example,  the  sum  of  two 
sects  is  a  sect,  which  may  be  found  directly  by  placing  the 
given  sects  end  to  end  in  a  straight,  with  none  but  these 
end-points  in  common.  The  sect  between  the  non-coinci- 
dent end-points  is  the  sum.  But  in  calculation  we  add  the 
lengths  (i.e.,  the  numbers  which  are  the  ratios  of  the  sects 
to  any  unit-sect)  of  the  sects,  and  obtain  the  length  of  their 
sum  (i.e.,  the  number  which  is  the  ratio  of  the  sum-sect  to 
the  same  unit). 

211.  As  stated  in  Section  203,  the  measurement  of  a 
magnitude  consists  in  finding  its  ratio  to  another  magni- 
tude of  the  same  kind,  chosen  as  a  basis  of  comparison. 
Howsoever  this  ratio  may  be  found,  the  magnitude  is 
measxwed. 

In  physical  science  magnitudes  are  commonly  measured, 
not  directly,  but  indirectly;  that  is,  the  direct  comparison 
is  not  between  the  magnitude  which  is  to  be  measured  and 
a  chosen  unit,  but  between  two  magnitudes  of  a  different 
kind  which  are  proportional  to  the  magnitude  which  is  to 
be  measured  and  its  unit.  It  is  highly  important  that  this 
fact  be  recognized  by  all  students  of  physical  science.  It 
also  emphasizes  very  clearly  the  absurdity  of  omitting  a 
sound  exposition  of  the  doctrine  of  proportionality  from 
elementary  instruction  in  mathematics. 

The  doctrine  of  proportionality  is  not  especially  difficult 
or  recondite ;  but,  even  if  it  were,  its  thorough  exposition 
cannot  be  postponed,  because  comprehension  thereof  is  pre- 
requisite for  understanding  ordinary  measurements  in  the 
most  elementary  physical  science,  and  the  commonplace 
problems  of    daily  life.     For  example,   temperatures   are 


128  NUMBER   AND   ITS   ALGEBRA. 

never  measured  directly,  but  always  by  raeaus  of  tlieir 
assumed  proportionality  to  the  volumes  of  some  body  at 
the  temperatures  in  question.  Again,  masses  are  usually 
measured  by  their  proportionality  to  the  corresponding 
weights  in  the  same  place,  etc. 

212.  Arcs  of  a  circle  are  so  conveniently  measured  by 
means  of  their  proportionality  to  the  angles  they  subtend 
when  the  vertices  of  the  angles  are  at  the  centre  of  the 
circle,  that  they  are  seldom  measured  directly.  It  must 
be  carefully  noted,  also,  that  only  angles  less  than  a  peri- 
gon  are  so  proportional,  and  therefore  so  measurable.  The 
indirectness  of  such  measurement  of  arcs  is  not  sufficiently 
emphasized  in  many  text-books.  The  most  faithful  Eng- 
lish translator  of  Euclid  long  ago  warned  teachers  of  the 
dangers  lurking  around  this  question.  In  the  first  of  his 
introductory  Dissertations,  he  gives  good  advice  for  leading 
a  pupil  to  attain  an  exact  and  adequate  concept  of  an 
angle,  and  especially  deprecates  any  association  of  angles 
and  arcs,  averring  himself  at  this  stage  "  afraid  to  meddle 
with  circular  arches,  lest  we  should  conjure  up  a  prejudice 
which  we  might  want  art  afterwards  to  layP 

In  more  than  one  instance  —  old  Roger  Ascham's  sage 
counsel  anent  teaching  Latin  comes  to  mind  —  modern 
tyros  in  pedagogics  would  have  done  better  to  consult  wise 
predecessors  than  to  follow  every  fad  of  educational  milli- 
ners as  they  vie  with  each  other  in  designing  latest  fash- 
ions. In  many  of  our  high  schools  it  would  be  difficult  to 
find  a  pupil  who  knows  exactly  what  an  angle  is  ;  and  not 
impossible  to  find  some  who  would  speak  of  an  arc  as 
equal  to,  or  half  of,  an  angle. 

213.  Definition.  —  One  series  of  magnitudes  of  the 
same  kind  is  proportional  to  another  series  of  magnitudes 


CRITERIA   OF   PROPORTIONALITY.  129 

of  the  same  or  of  a  different  kind,  corresponding  one-to-one 
to  the  first  series,  when  the  ratio  of  any  two  of  the  one 
series  equals  the  ratio  of  the  corresponding  two  of  the 
other  series. 

Tliis  is  the  direct  meaning  of  the  statement  tliat  one 
series  of  magnitudes  is  proportional  to  another  series. 
Criteria  sufB.cient  to  prove  this  relation  Avill  be  discussed 
presently. 

214.  Too  much  prominence  is  commonly  given  to  the 
case  where  each  series  consists  of  two  magnitudes.  Of 
course  two  magnitudes  are  proportional  to  two  others, 
when  a  ratio  of  the  one  pair  equals  the  corresponding  ratio 
of  the  other  pair. 

215.  Criteria  sufficient  to  prove  the  relation  of  propor- 
tionality are  often,  and  on  high  authority,  set  forth  as  defi- 
nitions of  proportionality.  Of  course  there  is  no  error  in 
this  ;  but  it  appears  to  create  confusion.  I  believe  it  is 
the  principal  explanation  of  the  not  uncommon  opinion 
that  the  true  doctrine  of  proportionality  is  unteachable  to 
high-school  pupils. 

216.  Euclid's  definition  of  equality  of  ratios  affords  the 
usual  criterion  of  proportionality :  Two  series  of  magni- 
tudes will  be  proportional  provided  that,  if  any  equimulti- 
ples of  a  corresponding  pair  of  magnitudes  one  in  each 
series  be  taken,  and  any  equimultiples  whatsoever  of  any 
other  corresponding  pair  be  taken,  then  the  multiple  of  the 
first  magnitude  in  one  series  is  greater  than,  equal  to,  or 
less  than  the  multiple  of  the  second  of  the  same  series, 
according  as  the  multiple  of  the  first  taken  of  the  other 
series  is  greater  than,  equal  to,  or  less  than  the  multiple 
of  the  second  of  that  series. 

217.  That  these  requirements  are  capable  of  being  ap- 


130  NUMBEE,   AND   ITS   ALGEBRA. 

plied  as  a  test,  may  be  shown  by  the  following  case :  Rec- 
tangles of  equal  bases  are  proportional  to  their  altitudes 
upon  those  bases. 

Here  a  series  of  surfaces  (vide  §  229),  corresponding  one- 
to-one  to  a  series  of  lines,  is  declared  proportional  to  the 
latter.  It  is  so,  for  the  ratio  of  any  two  of  the  surfaces 
equals  the  ratio  of  the  corresponding  two  of  the  lines ;  be- 
cause, if  upon  sects  equal  to  the  given  base  two  rectangles 
be  constructed  whose  altitudes  are  any  multiples  of  the 
altitudes  of  any  two  of  the  given  rectangles,  the  rectangles 
so  formed  are  respectively  the  same  multiples  of  the  ori- 
ginal rectangles.  Thus  equimultiples  of  a  corresponding 
pair,  one  in  each  series,  and  any  equimultiples  of  a  second 
corresponding  pair,  have  been  taken.  Also,  if  the  altitude 
of  one  of  these  trial  rectangles  be  greater  than  the  altitude 
of  the  other,  the  first  rectangle  is  greater  than  the  second ; 
and  if  equal,  equal ;  and  if  less,  less.  Therefore,  the  ratio 
of  any  two  of  the  series  of  rectangles  equals  the  ratio  of 
the  corresponding  two  of  the  altitudes ;  that  is  to  say,  the 
rectangles  are  proportional  to  the  altitudes. 

Note  that  all  this  is  regardless  of  commensurability  of 
the  altitudes  or  of  the  rectangles. 

218.  Euclid's  criterion  has  been  objected  to  because  it 
is  required  that  the  conditions  be  satisfied  for  any,  that  is 
all,  multiples ;  and  it  is  impossible  to  try  all  primary  num- 
bers. This  objection  is  not  valid,  though  there  may  be 
cases  which  require  a  searching  test  in  order  to  avoid  error. 
For  example  :  Consider  the  numbers,  4  and  3,  and  5  and  4. 

]\Iultiplying  the  two  antecedents  each  by  6,  and  the  two 
consequents  each  by  9,  we  get  24,  27 ;  30,  36  —  where 
24  <  27,  and  30  <  36.  Making  multiples  in  like  manner 
with  6  and  7,  we  get  24,  21 ;  30,  28  —  where  24  >  21,  and 


CRITEIIIA   OF   PROPORTIONALITY.  131 

SO  >  28.  ISTevertheless,  4  and  3  are  not  proportional  to 
5  and  4.  Thus  we  see  that  the  criterion  may  be  satisfied 
for  certain  multiples,  and  yet  not  satisfied ;  for  it  demands 
that  the  excess  or  defect  be  on  the  same  side  for  all  mul- 
tiples under  the  stated  conditions.  In  the  example  cited, 
if  Ave  use  10  and  13  for  multipliers,  Ave  get  40,  39 ;  50, 
52  — Avhere  40  >  39,  but  50  <  52. 

Of  course,  Avhere  the  question  concerns  the  jDroportion- 
ality  of  four  given  numbers  there  is  no  occasion  to  apply 
any  general  criterion  ;  but  the  relation  may  be  tested  im- 
mediately by  a  comparison  of  the  ratios.  Thus,  if  4,  3  ;  5, 
4  are  proportional  4/3  equals  5/4;  but  4/3  does  not 
equal  5/4,  because  4/3  -  5/4  =  16/12  -  15/12  =  1/12. 

219.  Alternative  criteria,  especially  adapted  to  test  pro- 
portionality in  many  cases  Avhich  arise  in  geometry  and 
physics,  are  presented  and  their  adequacy  established,  on 
page  93  of  Halsted's  Synthetic  Geometry  (John  Wiley  and 
Sons,  NcAV  York)  :  — 

Tavo  series  of  magnitudes  Avhicli  correspond  one-to-one, 
are  proportional  (that  is  to  say,  the  ratio  of  any  tAvo  of  the 
first  series  equals*  the  ratio  of  the  corresponding  two  of 
the  second  series)  provided  (1)  If  any  tAvo  of  the  one 
series  are  equal,  so  are  the  corresponding  two  of  the  other 
series  ;  and  (2)  To  the  sum  of  any  tAvo  of  the  One  series 
corresponds  the  sum  of  the  corresponding  tAvo  of  the  other 
series. 

For  example  : — The  intercepts  made  by  a  system  of 
parallel  straights  upon  one  transversal  are  proportional  to 
the  intercepts  made  upon  any  other  transversal :  for  if  any 
two  intercepts  on  one   transversal  are  equal,  so  also  are 

*  Fide  Section  83  (6). 


132  NUMBER    AND   ITS   ALGEBRA. 

the  corresponding  two  on  another  transversal ;  and  to  the 
sum  of  two  on  one  transversal  corresponds  the  sum  of 
the  corresponding  two  on  the  other  transversal. 

Again,  consider  arcs  of  the  same  or  equal  circles  and 
their  chords.  Arcs  are  not  proportional  to  their  chords, 
because,  although  if  two  of  the  arcs  are  equal  their  chords 
are  equal,  yet  to  the  sum  of  two  arcs  does  not  correspond 
the  sum  of  their  chords. 

220.  For  the  continuous  magnitude,  Space,  the  scientific 
fundamental  unit  is  the  metre,  which  is  the  sect  between 
two  marks  on  a  metal  bar  preserved  at  Paris.  The  sect 
is  to  be  taken  when  the  bar  is  at  the  temperature  of  melt- 
ing ice.  This  temperature  has  the  advantage  of  being 
readily  fixed  ;  but  a  point  so  far  from  ordinary  working 
temperatures  requires  the  correction  of  all  observations  of 
objects  not  iced,  and  coefficients  of  expansion  need  to  be 
accurately  known  for  all  substances  employed.  The  origi- 
nal (1799)  French  standard  metre  is  a  platinum  bar 
end-standard  about  1  inch  wide  and  i  inch  thick.  End- 
standards  are  objectionable  because  they  can  be  observed 
only  by  contact,  and  attrition  at  the  ends  is  inevitable. 
The  new  standard  of  the  International  Metric  Commission 
is  a  line-standard  of  platino-iridium,  about  40  inches  long 
and  0.8  inch  square,  grooved  on  four  sides  so  that  its 
section  is  between  an  X  and  H  form.  This  gives  rigidity 
and  a  surface  in  the  axis  of  the  bar  to  bear  the  lines  of  the 
standard. 

This  standard  is  preserved  at  the  International  IMetric 
Bureau  at  Paris,  where  the  most  refined  methods  of  com- 
parison are  provided  for,  and  which  is  supported  and  di- 
rected by  seventeen  nations. 

The  legal  theory  of  the  Metric  System  of  Units  is  :  — 


PHYSICAL   UNITS.  133 

(1)  The  standard  metre,  with  decimal  fractious  and  mul- 
tiples thereof.  (2)  The  litre  (declared  to  be  a  cube  of  0.1 
metre  edge),  with  decimal  fractions  and  multiples.  (3) 
The  kilogram  (defined  as  the  weight  in  vacuum  of  a  litre 
of  water  at  4°C.),  with  decimal  fractions  and  multiples. 

No  standard  litre  exists,  all  liquid  measures  being  fixed 
by  weight. 

When  established  in  1799  the  metre  was  supposed  to  be 
one  ten-millionth  of  the  terrestrial  quadrant  through  Paris. 
It  differs  from  this  fanciful  value  by  about  ^oVo- 

The  merits  of  the  metric  system  of  units  were  briefly 
discussed  in  Section  31. 

221.  The  fundamental  units  for  the  measurement  of 
physical  magnitudes,  chosen  by  the  Units  Committee  of 
the  British  Association,  and  unquestionably  the  most  sci- 
entific ever  agreed  upon,  are  the  centimetre,  gram,  and 
second.  The  system  is  known  as  the  C.G.S.  sj^stem.  For 
details  of  its  application  to  all  branches  of  physical  sci- 
ence (e.g.,  to  electricity)  the  student  is  referred  to  Pro- 
fessor Everett's  Units  and  Physical  Constants,  Macmillan 
and  Co. 


134  ^*u:MBER  a>'d  its  algebra. 

XIV.     ^Mathematics 

222.  rormal  thouglit,  consciously  recognized  as  such,  is 
the  means  of  all  exact  knowledge ;  and  a  correct  under- 
standing of  the  main  formal  sciences,  Logic  and  !Mathe- 
maticSj  is  the  proper  and  only  safe  fovmdation  for  a  scientific 
education. 

The  origin  and  nature  of  the  truths  of  the  formal  sci- 
ences are  not  so  recondite  as  they  are  often  made  to  appear. 
The  validity  of  Eeason  is  the  sole  postulate.  Mathematical 
truths  are  discovered  as  the  results  of  rational  operations 
upon  certain  elementary  concepts  determined  by  the  defini- 
tions with  which  the  science  begins.  The  operations  are 
not  capricious,  nor  is  their  nature  arbitrary.  They  are  not 
empty  words,  but  realities  —  not  '•  material "'  realities,  but 
all  the  more  real.  For  example,  numbers,  as  we  have  seen, 
are  not  concrete  things ;  and  as  soon  as  we  forget  that  they 
are  the  products  of  rational  processes,  we  at  once  fall  into 
error  and  confusion.  Such  confusion  is  most  prominent 
in  concepts  of  zero  and  infinity.  A  vague  concreting  of 
infinity  is  often  observable,  even  among  those  who  do  not 
make  a  like  mistake  with  any  other  number.  Because  In- 
finity as  a  concrete  is  inconceivable,  the  number  infinity 
is  commonly  spoken  of  as  inconceivable,  and  a  prevalent 
opinion  regards  finite  numbers  as  the  only  ones  we  can 
reason  about.  Charles  S.  Pierce,  eminent  as  logician  and 
mathematician  (and  mastery  of  both  sciences  is  requisite 
to  authority  in  either),  says,  "I  long  ago  showed  that  finite 
collections  are  distinguished  from  infinite  ones  only  by  one 
circumstance  and  its  consequences ;  namely,  that  to  them 
{the  finite)  is  applicable  a  peculiar  and  unusual  mode  of 
reasoning  called  by  its  discoverer,  DeMorgan,  the  'syllo- 


IKFlNItY.  135 

gism  of  transposed  quantity.'  .  .  .  DelVIorgan,  as  an  actu- 
ary, might  have  argued  that  if  an  insurance  company  pays 
to  its  insured  on  an  average  more  than  they  have  ever  paid 
it,  including  interest,  it  must  lose  money.  But  every  mod- 
ern actuary  would  see  a  fallacy  in  that,  since  the  business 
is  continually  on  the  increase.  But  should  war,  or  other 
cataclysm,  cause  the  class  of  insured  to  be  a  finite  one,  the 
conclusion  would  turn  out  painfully  correct  after  all.  .  .  . 
If  a  person  does  not  know  how  to  reason  logically,  and 
I  must  say  that  a  great  many  fairly  good  mathematicians 
—  yea,  distinguished  ones  —  fall  under  this  category,  but 
simply  uses  a  rule  of  thumb  in  blindly  drawing  inferences 
like  other  inferences  that  have  turned  out  well,  he  will, 
of  course,  be  continually  falling  into  error  about  infinite 
numbers.  The  truth  is,  such  people  do  not  reason  at  all. 
But  for  the  few  who  do  reason,  reasoning  about  infinite 
numbers  is  easier  than  about  finite  numbers."  * 

In  regard  to  infinitesimals  (the  word  is  simply  the  Latin 
ordinal  form  of  infinity),  and  contending  opinions  concern- 
ing the  methods  of  the  Infinitesimal  Calculus,  it  may  be 
remarked  that,  under  the  true  doctrine  of  continuity  and 
limits,  infinitesimals  are  presupposed,  and  that  there  can 
be  no  reason  except  expediency  to  shun  them  in  the  differ- 
ential calculus.  And  since  they  are  indispensable  for  the 
integral  calculus,  Mr.  Pierce  is  probably  right  in  his  view 
of  the  proper  procedure  of  the  Avhole  discipline,  when  he 
says,  in  the  paper  quoted  above,  '■'■  as  a  mathematician,  I 
prefer  the  method  of  infinitesimals  to  that  of  limits,  as 
far  easier  and  less  infested  with  snares."  At  all  events^ 
any  avoidance  of  infinitesimals  as  absurdities,  or  as  offer- 

*  "Law  of  Mind,"  Monht,  July,  1892. 


136  NUMBER   AND   ITS   ALGEBRA. 

ing  obstacles  to  sound  and  lucid  reasoning,  is  unneces- 
sary. 

223.  Mathematics  has  often  been  characterized  as  the 
most  conservative  of  all  sciences.  This  is  true  in  the 
sense  of  the  immediate  dependence  of  new  upon  old  re- 
sults. All  the  marvellous  new  advancements  presuppose 
the  old  as  indispensable  steps  in  the  ladder.  It  is  on  this 
account  that  "  there  is  no  royal  road "  to  mathematics. 
This  inaccessibility  of  special  fields  of  mathematics,  ex- 
cept by  the  regular  way  of  logically  antecedent  acquire- 
ments, renders  the  study  discouraging  or  hateful  to  weak 
or  indolent  minds.  In  reality  similar  demands  are  made 
by  every  science  ;  but  elsewbere  they  are  not  so  imperious, 
so  uncompromising.  It  is  possible  for  one  who  has  not 
mastered  fundamental  knowledge  in  the  sciences  of  phi- 
lology, history,  biology,  physics,  chemistry,  etc.,  to  nurse 
the  delusion  of  proficiency  and  comprehension  of  advanced 
problems ;  but  mathematics  is  inviolable  against  such  vain 
assaults.  Instant  and  conscious  is  the  curb  upon  her  vota- 
ries of  inadequate  knowledge. 

The  modern  tendency  to  dangerously  narrow  specializa- 
tion within  the  bounds  of  one  science  is,  also,  more  surely 
checked  in  mathematics  than  elsewhere.  The  attempt 
was  made  in  mathematics  as  in  the  other  sciences ;  but  it 
has  been  restrained.  Professor  Felix  Klein  remarked  at 
the  opening  of  the  Mathematical  and  Astronomical  Con- 
gress at  Chicago,  in  1893  :  "  When  we  contemplate  the  de- 
velopment of  mathematics  in  this  nineteenth  centur}^,  we 
find  something  similar  to  what  has  taken  place  in  other 
sciences.  The  famous  investigators  of  the  preceding  pe- 
riod were  all  great  enough,  to  embrace  all  branches  of 
mathematics.  .  .  .     With  the  succeeding  generation,  how- 


MATHEMATICS.  137 

ever,  tlie  tendency  to  specialization  manifests  itself.  .  .  . 
Such  conditions  are  unquestionably  to  be  regretted.  .  .  . 
I  wish  on  the  present  occasion  to  state  and  to  emphasize 
that  in  the  last  two  decades  a  marked  improvement  from 
within  has  asserted  itself  in  our  science  with  constantly 
increasing  success.  The  matter  has  been  found  simpler 
than  was  at  first  believed.  It  appears  indeed  that  the  dif- 
ferent branches  of  mathematics  have  actually  developed  — 
not  in  opposite,  but  in  parallel  directions,  that  it  is  possible 
to  combine  their  results  into  certain  general  conceptions. 
...  A  distinction  between  the  present  and  the  earlier 
period  lies  evidently  in  this  :  that  what  was  formerly 
begun  by  a  single  master  mind,  we  now  must  seek  to  ac- 
complish by  united  efforts  and  co-operation." 

224.  Another  trait  of  mathematics  which  renders  it  at- 
tractive to  some  minds  and  repellant  to  others,  is  its  self- 
sufficiency,  its  isolation,  its  independence  of  other  sciences. 
But  it  must  never  be  forgotten  that  mathematics  is  ever  at 
the  service  of  other  sciences ;  and  it  is  for  them  to  so 
formulate  their  problems  as  to  make  them  susceptible  of 
mathematical  treatment.  Indeed,  in  some  instances,  the 
difficulties  which  balk  a  thorough  investigation  of  certain 
physical  phenomena  consist  in  the  mathematical  problems 
encountered  in  the  solution  of  numerical  equations,  the 
summation  of  numerical  series,  etc.,  which  the  skill  of 
experimenters  has  succeeded  in  deducing  from  the  phenom- 
ena in  question.  Thus,  on  the  one  hand,  the  physicist  or 
economist  is  unceasingly  occupied  in  attempting  to  express 
the  relations  of  the  entities  with  which  he  deals,  as  some 
numerical  function  known  to  be  within  the  reach  of  mathe- 
matical reduction,  —  for  so  compendious  is  the  language  of 
Algebra,  that   theoretically  most   quantitative  and   many 


138  NUMBER   AND   ITS    ALGEBRA. 

qualitative  relations  are  somehow  expressible  as  numerical 
relations.  On  the  other  hand,  the  algebraist  is  constantly 
striving  to  bring  more  and  more  algebraic  forms  within  the 
powers  of  his  analysis.  By  their  joint  labors  the  confines 
of  knowledge  are  steadily  widened. 

225.  It  may  be  helpful  to  offer  a  definition  of  INIathe- 
matics,  not  in  the  sense  of  final  delimitation,  but  in  order 
to  afford  a  clear  notion  of  what  is  meant  by  subjects  or 
relations  capahle  of  viathematlcal  treatment.  I  cannot  do 
better  than  quote  Professor  George  Chrystal  in  his  article 
on  "  Mathematics,"  Encydopcedia  Britannica,  ninth  edi- 
tion, who  makes  the  f  olloAving  definition  :  "  Any  concept  * 
which  is  definitely  and  completely  determined  by  means 
of  a  finite  number  of  specifications,  say  by  assigning  a 
finite  number  of  elements,  is  a  mathematical  concept. 
Mathematics  has  for  its  function  to  develop  the  conse- 
quences involved  in  the  definition  of  a  group  of  mathe- 
matical concepts.  Interdependence  and  mutual  logical 
consistency  among  the  members  of  the  group  are  postu- 
lated, otherwise  the  group  would  either  have  to  be  treated 
as  several  distinct  groups,  or  would  lie  beyond  the  sphere 
of  mathematics." 

226.  Examples  of  concepts  completely  determined  by  a 
finite  number  of  specifications  are  familiar.  On  the  other 
hand,  horse,  tree,  gold,  becmty,  love,  are  examples  of  non- 
mathematical  concepts.  Of  course,  Kumber  may  be  ab- 
stracted from  these,  or  any  other  separate  objects  of 
thought  or  sense-perception,  and  Number  is  a  mathemati- 
cal concept ;  but  the  concept  of  a  number  of  trees  is  not  at 
all  the  concept  of  the  trees.     Again,  the  form  of  an  irregu- 

*  I  have  taken  the  liberty  of  changing  the  word  "  conception  "  to 
concept  three  times  in  this  passage. 


ELEMENTAL   CONCEPTS.  139 

lar  piece  of  wood  canuot  be  determined  by  a  finite  number 
of  specifications,  and  its  form  therefore  cannot  be  mathe- 
matically treated  (its  weight  of  course  could).  But  if  from 
this  irregular  piece  of  wood  a  sphere  be  turned,  its  form 
is  specified  by  stating  that  it  is  a  sphere,  and  giving  the 
length  of  its  radius.  This  illustrates  at  once  the  bounda- 
ries of  mathematics,  and  the  relation  of  mathematics  to 
the  arts. 

227.  ]Mensuration  is  an  important  function  of  mathemat- 
ics ;  but  it  occupies  too  prominent  a  place  in  some  notions 
of  the  subject-matter  of  the  science.  I  have  already  ((7f. 
Introduction,  p.  13,  and  §  12)  referred  to  the  mistake  of 
assigning  the  origin  of  Number  to  measurement.  Nor  is 
the  prevalent  notion  that  mathematics  is  the  '^  science  of 
quantity  "  correct.  Projective  geometry  in  the  purity  of 
its  recent  development  is  displayed  as  a  mathematical 
treatment  and  method  well-nigh  void  of  quantitative  rela- 
tions, and  dealing  for  the  most  part  with  qualitative  re- 
lations of  spacial  manifoldnesses.* 

228.  When  we  reach  elementary  concepts  we  always 
find  that  they  cannot  be  defined  except  in  cognate  terms. 
Such  elemental  concepts  are  quality,  one,  many,  space,  time, 
and  the  interrelated  concepts,  whole,  part,  more,  less,  equal, 
quantity.  All  that  can  be  done  in  the  way  of  defining  such 
concepts,  is  to  exhibit  the  phenomena  from  which  they 
have  been  abstracted,  and  the  processes  of  abstraction ; 
and  then,  for  purposes  of  exact  expression,  make  the  def- 
inition in  cognate  terms.  A  good,  short  definition  of 
quantity  according  to  a  standard  dictionary  (the  Century) 
is :  "  The  intrinsic  mode  by  virtue  of  which  a  thing  is  more 

*  See  a  work  on  projective  geometry  by  Dr.  Halsted,  just  now  in 
press. 


140  NU5IBER   AND   ITS   ALGEBRA. 

or  less  than  another."  {Mode  =  system  of  relationship.) 
It  is  plain  that  some  things  exist  in  this  mode;  that  is, 
possess  quantity,  which  are  not  magnitudes,  or  manifold- 
nesses,  in  the  mathematical  sense.  For  example,  beauty  is 
quantitative,  is  more  or  less  ;  but  in  no  proper  sense  can  it 
be  added  to  itself  so  as  to  double.  It  is  a  loose  figure  of 
speech  to  say,  the  beauty  of  one  thing  is  tAvice  that  of 
another,  as  is  at  once  apparent,  should  we  go  on  to  say 
that  it  was  eleven  times  that  of  some  other. 

229.  ]\Iany  words  have  been  used  to  denote  the  charac- 
teristic relation  of  a  mathematical  concept  to  its  elements. 
Magnitude  and  quantity  are  the  familiar  terms.  In  the 
preceding  discourse  I  have  employed  the  former  term  ;  but 
for  reasons  both  of  intrinsic  propriety,  and  less  ambiguity 
owing  to  irregular  usage,  the  word  manifoldness,  which 
has  lately  come  into  use,*  is  perhaps  the  most  fitting 
term ;  though  manifoldness  is  also  used  to  denote  a  group 
of  correlated  magnitudes  differing  in  kind. 

Quantity  and  magnitude  are  each  used  in  two  respec- 
tively synonymous  senses.  Either  magnitude  or  quantity 
may  be  found  defined  for  mathematical  purposes  as  ''  any- 
thing which  may  be  added  to  itself  so  as  to  double ;  "  and 
yet  the  same  writer  may  be  found  speaking  of  the  magni- 
tude of  some  such  magnitude,  or  the  quantity  of  some 
such  QUANTITY.  The  ancient,  and  still  universally  cur- 
rent, categorical  sense  of  quantity  seems  to  me  to  render 
it  the  more  appropriate  terra  for  the  sense  of  the  italicized 
words  in  the  phrases  cited ;  and  therefore,  by  exclusion, 


*  Cf.  the  article  on  "  :Matliematics"  by  Prof.  Chrj'stal,  above  referred 
to;  also  the  article  on  "Measurement"  by  Sir  Robert  Ball,  Royal  As- 
tronomer for  Ireland. 


MANIFOLDNESS.  141 

magnitude  should  be  confined  to  the  sense  of  the  capi- 
talized words. 

Manifoldness  in  one  sense  is  entirely  synonymous  with 
magnitude  in  the  use  I  have  made  of  the  latter.  But  be- 
sides a  single  totality,  manifoldness  often  means  a  single 
system  of  different  totalities,  and  the  difference  may  be  in 
kind.  Thus,  a  line  is  a  manifoldness,  so  is  a  surface,  so  is 
an  angle ;  yet  the  system  of  lines,  along  with  the  angles 
and  surface  determined,  which  we  call  a  triangle,  is  also 
termed  a  manifoldness. 

Now,  a  triangle,  in  the  sense  of  the  whole  figure,  is  not  a 
magnitude ;  its  surface  is  a  magnitude,  its  sides  are  magni- 
tudes, and  its  angles  are  magnitudes.  The  word  triangle 
often  plainly  means  exclusively  the  surface  of  the  triangle, 
and  the  abbreviation  is  legitimate  where  there  is  no  danger 
of  confusion  ((7/*.  §  217)  ;  but  if  triangle  means  the  entire 
definite  system  of  surface,  lines,  and  angles,  then,  clearly, 
a  triangle  is  not  a  magnitude,  but  a  system  of  different 
magnitudes.  But  a  triangle,  in  this  sense  of  the  whole 
figure,  the  system  of  magnitudes,  three  sects,  three  angles, 
and  one  surface,  is  called  a  manifoldness.  Such  manifold- 
nesses  have  been  termed  discrete ;  but  this  is  a  totally  dif- 
ferent sense  of  discrete,  from  its  meaning  in  any  statement 
that  a  single  magnitude  is  discrete,  e.g.,  Primary  Number 
is  a  discrete  magnitude.  Discrete  is  the  antithesis  or  an- 
tonym of  co7itimious.  Most  magnitudes  are  continuous ; 
number,  time,  and  space  are  the  great  continuums,  with 
which  mathematics  has  most  to  do.  If  mayiifoldness  is  to 
be  used  in  this  double  sense,  it  is  necessary  to  distinguish 
the  meanings  by  some  adjectives ;  and  discrete  is  not  a 
good  term  for  the  latter  sense.  Homogeneous  and  dis- 
parate would  not  be  abusive  terms.     I  shall  use  them. 


142  NUMBER   AND   ITS    ALGEBRA. 

230.  Xumber  is  tlie  very  web  of  mathematics,  the  mani- 
foldness  iipon  which  are  woven  investigations  concerning 
all  other  manifoldnesses  whatsoever.  All  other  manifold- 
nesses  are  even  fundamentally  determined  (as  will  pres- 
ently appear)  by  means  of  Xumber ;  but  Number  determines 
itself. 

Geometry  cannot  even  apparently  proceed  without  arith- 
metic. Euclid  makes  the  formal  connection  in  his  fifth 
book ;  but  there  is  a  more  primary  and  essential  connec- 
tion. We  have  considered  the  error  of  seeking  geometric 
definitions  of  number,  particularly  negative,  neomonic,  and 
complex  number.  But  the  tables  are  entirely  turned  Avhen 
we  consider  that  geometric  or  any  other  manifoldnesses  are 
defined  in  some  very  fundamental  properties  by  means  of 
number. 

231.  Most  text-books  on  stereometry  set  forth  that  all 
solids  have  three  dimensions,  length,  breadth,  and  thick- 
ness. But  what  does  this  exactly  mean  ?  What  is  the 
length,  breadth,  and  thickness  of  a  pyramid,  a  rough  stone, 
a  bunch  of  grapes  ?  Xo  solids,  except  cubes  or  right  paral- 
lelopij)eds,  clearly  determine  three  principal  directions  in 
which  length,  breadth,  and  thickness  may  be  discerned. 

The  dimensions  are  clearly  and  sharply  defined  only  by 
considering  the  number  of  specifications  necessary  and  suf- 
ficient to  fully  determine  any  element.  Thus,  solid  space 
regarded  as  point  aggregates  is  tri-dimensional,  because, 
given  three  concurrent  straights  or  planes,  as  ground  of 
reference,  three  numbers  are  necessary  and  sufficient  to 
determine  any  one  point-element,  distinguishing  it  from  all 
others. 

Note  also  that  the  space  of  our  experience  is  four-dimen- 
sional  if  regarded   as   an   assemblage  of   geodesic   lines, 


MANIFOLDNESSES.  l-i3 

because  in  that  case  four  numbers  are  required  to  deter- 
mine one  element. 

232.  Manifoldnesses,  homogeneous  or  disparate,  are  one- 
dimensional,  two-dimensional,  etc.,  (or  one-fold,  two-fold, 
etc.),  according  as  in  the  totality  or  system  considered,  one 
number,  or  two  numbers,  etc.,  are  necessary  and  sufficient 
to  determine  and  distinguish  any  particular  element  in  the 
homogeneous  totality,  or  in  the  system. 

The  distinction  between  homogeneous  and  disparate  mani- 
foldnesses must  not  be  confounded  with  that  between 
continuous  and  discrete  manifoldnesses.  A  homogeneous 
manifoldness  is  either  continuous  or  discrete ;  a  dispa- 
rate manifoldness  is  a  system  of  homogeneous  (continuous 
or  discrete)  manifoldnesses.  Disparate  denotes  a  system 
of  manifoldnesses  differing  in  kind ;  that  is,  such  as  could 
not  be  compared  with  one  another  {vide  §  207),  e.g.,  the 
surface,  lines,  and  angles  of  a  triangle.  As  already  said, 
most  homogeneous  manifoldnesses  are  continuous.  Pri- 
mary Number  is  the  conspicuous  discrete  magnitude  with 
which  we  have  to  do. 

According  to  different  standpoints,  the  same  manifold- 
ness may  be  of  various  dimensions. 

233.  Examples.  — A  straight  line  regardless  of  position, 
time,  temperature,  probability,  the  totality  of  all  spheres  dis- 
tinguished, not  in  respect  of  position,  but  solely  in  regard 
to  size  or  quantity,  are  one-fold  manifoldnesses.  All  such 
are  homogeneous,  for  of  course  no  one-fold  manifoldness 
could  be  disparate. 

The  assemblage  of  points  on  a  plane,  the  sphere  as  sur- 
face {Cf.  latitude  and  longitude),  are  two-fold  manifold- 
nesses. 

Space  as  an  assemblage  of   points    is  a  tri-dimensional 


144  NUMBER   AND   ITS   ALGEBRA. 

manifoldness.  A  triangle  considered  without  reference  to 
position'  (because  it  may  be  completely  determined  in  vari- 
ous ways  by  assigning  three  elements)  is  a  triple  disparate 
manifoldness. 

The  totality  of  all  spheres  each  to  be  completely  deter- 
mined is  a  four-fold  manifoldness. 

Since  a  plane  quadrilateral  is  completely  determined 
when  five  elements  are  known,  it  is  a  quintuple  or  five- 
fold disparate  manifoldness. 

A  plane  w-gon  in  like  manner  is  a  (2  ?i  —  3)-fold  dis- 
parate manifoldness. 

234.  There  are  two  general  methods  in  the  mathemati- 
cal investigation  of  manifoldnesses.  They  are  called  the 
synthetic,  or  synoptic  method,  and  the  analytic  method. 
The  analytic  method  is  mainly  numerical;  the  synthetic 
deals  directly  with  the  magnitudes  considered,  and  only 
unavoidable  numerical  relations  are  involved.  Of  course 
there  is  no  sharp  line  of  demarcation,  and  the  two  methods 
yield  identical  results. 

In  geometry  metrical  relations  are  in  general  more  readily 
investigated  by  the  analytic  ;  descriptive  properties  by  the 
synoptic  method. 

235.  The  synthetic  method  is  peculiarly  fitted  to  pure 
geometry,  but  this  is  not  its  only  field.  Ever  since  Rie- 
mann's  epoch-making  dissertation,  Ueher  die  Hypothesen 
u-elclte  (lev  Geometrie  zu  Grunde  Ueyen,  1854,  synoptic 
methods  have  been  applicable  to  w-fold  manifoldnesses ; 
and  the  applications  to  Statistics  and  Physics  are  familiar. 

236.  In  mathematics  all  analytic  methods  employ  an 
algebra  {vide  §  20  et  seq.)  ;  but  it  is  the  Algebra  of  Number 
which  is  the  most  highly  developed  and  powerful  instru- 
ment of  such  methods  of  research.     It  is  to  the  study  of 


ALGEBRAIC    FORM.  145 

this  organized  and  compendious  instrument  of  numerical 
expression  that  these  lectures  are  introductory.  Plainly 
the  first  step  to  the  understanding  of  the  algebra  of  num- 
ber is  to  understand  the  nature  and  laws  of  number.  It 
is  hoped  that  these  lectures  have  been  a  fairly  adequate 
guide  and  stimulus  to  this  step.  After  mastering  what 
may  be  called  the  vocabulary  of  the  language  (proficiency 
in  this  matter  has  been  assumed),  the  next  step  is  to 
grasp  the  idea  of  algebraic /b;v/i.  In  the  study  of  Algebra 
this  should  be  the  main  standpoint.  It  is  only  by  follow- 
ing out  the  problems  which  arise  in  a  systematic  study 
of  algebraic  form  tljat  the  modern  developments  of  pure 
algebra,  or  its  applications  to  geometry,  can  be  rightly 
comprehended. 

237.  In  conclusion,  I  may  say,  in  reference  both  to  this 
little  work,  and  to  any  text-book  which  may  engage  j^our 
attention,  that  if  a  mathematical  treatise  is  worth  reading 
at  all,  it  is  worth  re-reading,  and  reading  backwards  and 
forwards,  and  in  special  topics.  As  Professor  Chrystal 
says  in  the  preface  to  his  Text  Book  of  Algebra,  ''When 
you  come  on  a  hard  or  dreary  passage,  pass  it  over ;  and 
come  back  to  it  after  you  have  seen  its  importance  or 
found  the  need  for  it  further  on." 


"UNIVE 

C.4-..  '"' 


146  NUMBER   AND   ITS    ALGEBRA. 

XV.  Some  Theorems  axd  Problems. 

238.  Every  primary  number  is  a  multiple  (§  83)  of  one 
and  of  itself  :  if  it  has  no  other  submultiple,  it  is  called  a 
prime  number ;  if  it  has  another  submultiple,  it  is  called 
comjiosite. 

If  one  primary  number  is  a  submultiple  of  each  of  two 
or  more  others,  it  is  called  a  common  suhmidtij)le. 

Primary  numbers  (prime  or  composite)  with  no  common 
submultiple  other  than  unity,  are  said  to  be  prime  to  each 
other. 

239.  Theorem.  —  Every  composite  primary  number  can 
be  resolved  into  factors  which  are  positive  integral  powers 
of  prime  numbers. 

Every  primary  number  less  than  a  composite  number 
either  is,  or  is  not,  a  submultiple  of  the  latter :  let  a  be 
the  least  primary  number  (>  1),  that  is  a  submultiple  of 
the  composite  number,  A. 

Then  A  =  ax.  If  x  be  also  a  multiple  of  a,  x  =  oij,  and 
A  =  a^y.  Einall}'  A  =  a"^i(,  Avhere  u  is  either  1,  or  prime 
to  a,  and  either  prime,  or  a  multiple  of  some  prime  >  a 
and  <  A,  say,  b. 

In  like  manner  ti=h"v,  where  v<Cn-  and  v^cPw, 
where  to  <  v,  and  so  on. 

Clearly  the  process  must  end  with  1 ;  therefore 

A  =  aI"h'"cP  .  .   .  , 

where  a,  b,  c  .  .  .  are  prime  numbers. 

It  will  be  seen  below  that  this  resolution  can  be  effected 
in  only  one  way ;  also,  that  positive  integral  powers  of 
prime  numbers  are  prime  to  each  other. 

240.  Understanding  the  numerical  symbols  as  represent- 


HIGHEST    COMMON    SUBMULTIPLE.  147 

ing  integers  (positive  or  negative),  and  extending*  the 
meaning  of  the  term  multiple  to  include  the  relation  of 
one  number  to  another  if  the  ratio  of  the  former  to  the 
latter  be  integral,  then  :  — 

If  ^  is  a  multiple  of  a,  any  multiple  of  A  is  a  multiple  of 
a.     Tliis  is  obvious. 

Also,  if  A  and  B  have  a  common  submultiple,  m,  then 
Ax  -J-  By  is  a  multiple  of  m. 

For,  say,  A  =  pm.  and  B  =  qm  ; 

then  Ax  -|-  By  =  xpm  -\-  yqvi, 

and  therefore,  distributing  the  right-hand  member, 
Ax  -J-  By  =  m  (xp  +  yj). 

From  these  two  theorems  is  deduced  a  means  of  finding 
the  highest  f  common  submultiple  (h.  c.  s.)  of  two  or  more 
integers. 

For,  if  A  =  pB  -\-  c,  the  h.  c.  s.  of  A  and  B  is  the  h.  c.  s. 
of  B  and  c.  To  prove  this  it  is  necessary  and  sufficient  to 
show  — 

(1)  Every  submultiple  of  B  and  c  is  a  submultiple  of  A 
and  B. 

(2)  Every  submultiple  of  A  and  i?  is  a  submultiple  of  B 
and  c. 

(1)  As  just  shown,  every  submultiple  of  B  and  c  is 
a  submultiple  of  p) B  -\-  c,  that  is  of  A;  therefore,  every 
submultiple  of  B  and  c  is  a  submultiple  of  A  and  B. 

(2)  Since  A  =  pB  ^  c,  c  =  A  —  pB.  Therefore,  again, 
every  submultiple  of  A  and  i?  is  a  submultiple  of  ^  —  ^;  B, 

*  A  violent  extension  {Cf.  definition,  §  83) ;  but  custom  is  a  tyrant, 
and  brevity  tempting.  Some  such  term  as  co-multiple  would  adequately 
distingui-sh  this  relation,  e.g.,  of  12  to  —  3. 

t  The  term  highest  is  employed  in  order  to  avoid  contradictory  uses  of 
"  greatest"  in  regard  to  negative  numbers.     {Cf.  §  242.) 


148 


NUMBER   AND   ITS   ALGEBRA. 


that  is  of  c,  and  consequently  every  submultiple  of  A  and  B 
is  a  submultiple  of  B  and  c. 

Thus,  to  find  the  h.  c.  s.  of  A  and  B,  where  yl  >  J5,  we 
have,  by  successive  divisions,  — 

A  =  2>B  -\-  c  .  .  .  Avhere  c  <i  B, 


c  =  rd  -\-  e 


where  d  <i  c, 
where  e  <  rf. 


where  m  <  /, 

Avhere  n  <  ?»,  and  may  be  1, 
where,  if  ?i  =  1,  «•  =  vh. 
of  ^  and  B ;  for  the  h.  c.  s.  of  A 


Iz  =    vl  -\-  m 

I  =  vm  -\-  n 

and,  finally,      vi  =  loi 

Whence,  7i  is  the  h.  c.  s 
and  B  =  h.  c.  s.  of  B  and  e  =  h.  c.  s.  of  c  and  d  =  .  .  .  = 
h.  c.  s.  of  m  and  n.  But  m  =  wn,  and  therefore  7i  is  the 
h.  c.  s.,  since  n  can  have  no  submultiple  higher  than  itself. 

If  n  =  1,  A   and  B  have  no  common   submultiple  but 
unit}',  and  are  prime  to  each  other. 

For  example,  find  the  h.  c.  s.  of  A  =  2911  and  B  =  1763. 

The  calculation  may  be  compared  with  the  foregoing  as 

follows  :  — 

^  =  1763)2911  =.4(1  =i> 
1763 

0  =  1148)1763(1  =  q 


(^  =  615)  1148(1  =  r 
615 

e  =  533 )  615  ( 1  =  5 
533 
/*  =  82 )  533  (6  =  t 
492 


whence,  y  =  41  is  the  h.  c.  s. 


^  =  41)82(2  =  ?^ 
82 

0 


RELATIVE   PRIMENESS.  149 

It  must  be  discerned  that  the  essence  of  this  process  is 
merely  that  the  quotients  be  integral,  and  the  moduli  (^vide 
§  198)  of  the  dividends  be  in  decreasing  order,  for  qualita- 
tive distinctions  are  ignored ;  -[-  4,  for  instance,  being  in- 
differently the  h.  c.  s.  of  8  and  12. 

In  accordance  with  these  considerations  the  process  may 
be  abbreviated  in  various  ways.  If  convenient,  remainders 
may  be  negative,  and  any  submultiple  of  a  divisor  evi- 
dently prime  to  the  dividend,  or  submultiple  of  dividend 
prime  to  divisor,  may  be  cast  out.  The  above  calculation 
might  have  been  abbreviated  thus  :  — 


*&' 


Since  neither 
3  nor  5  is  a  sub- 
multiple  of  1763, 
15  may  be  cast 
out  of  615. 


1763)2911(2 
3526 

-  15)  -615 

41)1763(43 
164 

123 
123 


Every  common  submultiple  of  A,  B,  C  .  .  .  is  a  common 
submultiple  of  A  and  B,  and  therefore  of  vi,  the  li.  c.  s.  of 
A  and  B.  Consequently,  to  find  the  h.  c.  s.  oi  A,  B,  C  .  .  . , 
find  the  h.  c.  s.  of  m  and  C,  and  so  on. 

241.  It  follows  from  the  preceding  discussion,  that,  if  a 
and  b  be  prime  to  each  other,  any  common  submultiple  of 
aN  and  b  must  be  a  submultiple  of  N. 

Also,  if  a  be  a  submultiple  of  bN  and  prime  to  b,  it  is  a 
submultiple  of  N. 

Also,  if  a  be  prime  to  I,  m,  n  .  .  .,  it  is  prime  to  their 
product,  Imn ;  and  consequently  if  a,  b,  c  .  .  . ,  be  each 
prime  to  all  of  I,  vi,  n  .  .  . ,  the  product,  abc  .  .  . ,  is  prime 
to  the  product,  Imii  .... 


150  NUMBER   AND   ITS   ALGEBRA. 

In  particular,  if  a  be  prime  to  h,  «"  is  prime  to  Z/™.  This 
is  true,  of  course,  when  a  and  h  are  prime  numbers ;  that 
is  to  say,  positive  integral  powers  of  prime  numbers  are 
prime  to  each  other. 

Moreover,  an  integer  can  be  resolved  into  factors  which 
are  powers  of  prime  numbers  in  only  one  way.  {Vide 
§  239.) 

For,  if  two  resolutions  be  possible,  let  alcd  =  hnnrs. 

Then  ahcd  is  a  multiple  of  Z;  but  since  /  is  a  positive 
integral  power  of  a  prime  number,  it  is  prime  to  each  of  a, 
h,  c,  d,  except  one  which  is  a  not  less  power  of  the  same 
prime  number;  and  there  must  be  such  a  one,  or  I  could 
not  be  a  submultiple  of  cd>nl ;  —  say  a  is  this  one.  Again 
hnnrs  is  a  multiple  of  a,  and  it  follows  as  before  that  I 
must  be  a  multiple  of  a.  But  if  a  is  a  multiple  of  I, 
and  /  of  a,  a  =  I.  Likewise  three  more  of  vmis  must 
respectively  equal  h,  c,  and  d,  and  tlierefore  the  unpaired 
factor  must  be  1. 

242.  The  lowest  common  multiple*  of  two  integers  equals 
their  product  divided  by  their  highest  common  submultiple. 

For  if  A  =  sx  and  B  =  sy,  where  s  is  the  h.  c.  s.  of  A  and 
B,  then  AB  =  s^-xy.  But  s,  x,  and  y  are  prime  to  each 
other,  and  therefore  sxy  is  the  1.  c.  m.  of  A  and  B,  —  and 

AB 

sxy  = 


*  In  respect  to  primary  numbers,  the  term  least  common  multiple 
means  exactly  what  it  says;  hut  in  reference  to  both  positive  and  nega^ 
tive  integers  a  variance  in  the  meaning  of  the  term  "least"  is  to  he 
noted,  such  as  was  remarked  in  the  foot-note  of  Section  240  concerning 
"greatest."  Indifferent  alternatives— one  positive,  the  other  negative 
—  are  always  considered  in  the  highest  common  submultiple,  and  the 
lowest  common  multiple  of  two  integers. 


LOWEST    COMMON    MULTIPLE.  151 

Therefore,  to  find  the  lowest  common  multiple  of  two 
integers,  we  have  the  rules  :  — • 

Divide  their  product  by  their  h.  c.  s. ;  or 

Divide  either  by  their  h.  c.  s.,  and  multiply  the  other  by 
the  quotient ;  or 

Divide  each  by  their  h.  c.  s.,  and  take  the  product  of  the 
quotients  and  the  h.  c.  s. 

Any  one  of  these  three  rules  may  in  a  special  case  be  the 
most  convenient. 

The  1.  c.  m.  of  more  than  two  integers  is  the  1.  c.  m.  of 
the  1.  c.  m.  of  the  first  two  and  the  third,  and  so  on. 

243.  Plainly  (symbols  meaning  integers)  a  =  xh  -\-  r  in 
an  infinite  variety  of  ways  ;  for  x  may  be  fixed  arbitrarily 
and  r  found,  so  that  r  =  a  —  xh.  But  important  special 
cases  arise  if  (/,  h,  and  x  are  positive,  and  r  restricted :  — 

(1)  When  r  <  b. 

(2)  When,  though  r  is  negative,  mod  r  <  mod  h.  (Vide 
§  198.) 

In  both  cases  a  =  xb  -\-  r  in  only  one  yvciy. 

(1)  Tf  xb  be  the  greatest  multiple  of  b,  not  >  a,  then 
r  =  a  —  xb,  where  r  <  b.  Nor  could  there  be  a  second 
resolution  under  the  same  conditions,  else  xb  -f-  *'  would 
equal  x' b  -}-  r ',  and  therefore  r  —  r'  =  (x'  —  x)  b,  and  there- 
fore )•  —  r'  would  be  a  luultiple  of  b,  —  an  impossibility, 
since  r  and  r',  being  each  less  than  b,  r  —  r'  is  less  than  b. 

(2)  If  xb  be  the  least  multiple  of  b  not  <  a,  then 
a  —  xb  =  r,  where  r  is  negative,  but  mod  /■  <  mod  b ;  and 
the  resolution  is  unique  as  before. 

In  these  cases  r  is  called  the  least  positive  remainder 
and  "  least "  *  negative  remainder  of  a  with  respect  to  b. 

*  Cf.  foot-notes  to  Sections  240  and  242. 


152  NUMBER    AND   ITS   ALGEBRA. 

Least  remainder^  unqualified,  is  to  be  understood  in  the 
former  sense. 

Obviously  a  is  prime  to  h  if  the  least  remainder  of  a 
with  respect  to  h  does  not  vanish,  and  not  prime  if  it  does 
vanish. 

244.  Let  the  student  prove,  if  the  least  remainders  of  x 
and  y  with  respect  to  z  be  equal,  a;  —  ?/  is  a  multiple  of  z, 
and  inversely. 

245.  When  the  ratio  a;  /y  is  not  integral,  a-  /y  is  said  to 
be  essentially  fractional,  or  briefly,  fractional. 

li  a  jh  =  c  I  cl  when  ay  h  and  c  >  d,  prove  that  the  frac- 
tions, reduced  to  form  n  -{-  r/h,  where  r  <  h,  must  have 
their  integral  and  fractional  parts  equal  separately. 

246.  Pi^ove  :  11  A  [  B  =  a  /  h  and  a  /  h  is  at  its  lowest 
terms  (i.e.,  a  prime  to  h),  then  A  =  na  and  B  =  nh. 

247.  Prove  that,  using  only  positive  remainders  in  the 
process  of  finding  the  h.  c.  s.  of  two  positive  integers,  A  and 
B,  every  remainder  equals  i  {Ax  —  By),  where  x  and  y 
are  positive  integers,  and  the  upper  sign  goes  with  the  1st, 
3d,  etc.,  and  the  lower  with  the  2d,  4th,  etc.,  remainders. 

Also,  if  a  and  h  be  prime  to  each  other,  positive  integers 
can  always  be  found  such  that  xa  —  yh  =  J^l. 

It  is  obvious  that  these  numbers,  when  determined,  will 
be  prime  to  each  other,  for  by  Section  240,  1  is  a  multiple 
of  every  common  submultiple  of  x  and  y. 

248.  Prove : 

(1)  If  X  prime  to  y,  {x  +  y)"  and  {x  —  ?/)"  have  h.  c.  s. 

not  >  2". 

(2)  If  X  prime  to  y,  x"  +  ?/"  and  a;"  —  y"  are  prime,  or 

have  h.  c.  s.  =  2. 

(3)  If  X  prime  to  y,  a;  +  y  and  x^  -\-  y"^  —  oty  are  prime, 

or  have  h.  c,  s.  =  2  or  3. 


RADICAL   SURDS.  153 

(4)  The  difference  of  the  squares  of  two  odd  integers  is 

a  multiple  of  8. 

(5)  The   difference   of  the   squares  of  two  consecutive 

integers  equals  their  sum. 

(6)  The  product  of  three  consecutive  even  integers  is  a 

multiple  of  48. 

(7)  The  sum  of  the  squares  of  three  consecutive  odd 

numbers   and   1   is   a  multiple  of   12,  but  never 
of  24. 

(8)  The  product  of  the  cubes  of  three  consecutive  inte- 

gers is  a  multiple  of  their  sum. 

249.  At  several  points  in  preceding  chapters,  it  has  been 
taken  for  granted  that  the  operation  of  evolution  upon 
many  integers  results  in  essentially  surd  or  incommensu- 
rable number  ;  that  is  to  say,  that  no  fraction  can  possibly 
be  the  required  root  — •  although  fractions  approximating 
the  surd  as  nearly  as  desired  can  be  obtained.  Fractional 
number  is  still  discrete,  fractions  are  continuous  through 
surds.     (Vide  U  94,  Sl-82.) 

To  demonstrate  these  propositions,  it  is  enough  merely 
to  consider  that  no  power  of  an  essentially  fractional  num- 
ber can  be  an  integer.  For,  ii  x  /  y  is  a  fraction  in  its 
lowest  terms,  x  is  prime  to  i/,  and  therefore,  by  Section  241, 
any  power  of  x  is  prime  to  any  power  of  i/,  and  consequently 
any  power  of  a-/ ^  is  still  essentially  fractional. 

For  example  :  Obviously  no  integer  is  the  square  root 
of  7,  but  some  number  greater  than  2  and  less  than  3.  But 
this  number  is  no  fraction,  for,  as  just  shown,  no  power 
Avhatsoever  of  any  essentially  fractional  number  can  be  an 
integer.  Thus,  it  is  proved  that  the  familiar  process  of  ap- 
proximate calculation  of  roots  of  such  integers  is  absolutely 
interminable.       (Moreover,    the    endless   decimal   fraction 


154  NUMBER   AND   ITS   ALGEBRA. 

obtainable  can  never  form  a  repeating  period  of  figures  — 
(vide  §  284). 

In  tliis  Avay  it  is  plain  that  no  integers  except  the  series, 
1,  4,  9,  16  .  .  . ,  (the  squares  of  1,  2,  3,  4  .  .  . ,  and  called 
"  square  numbers ")  can  have  any  but  incommensurable 
square  roots ;  that  the  cube  roots  of  all  integers  but  1,  8, 
27,  64  .  .  .  (1^,  2%  3^,  4^  .  .  .)  are  incommensurable,  and 
so  on. 

250.  For  proof  of  the  proposition  :  The  number  of  prime 
integers  is  infinite  (see  Euclid,  IX,  20). 


251.  Attentive  perusal  of  the  following  sections  Avill 
bring  out  a  general  distinction  (correct  apprehension  of 
which  is  highly  important)  between  the  applications  of  a 
confusingly  similar  terminology  to  individual  numbers  and 
to  analytical  functions  of  such  numbers,  —  the  distinction 
between  algebraic  form,  and  particular  numerical  values. 

For  example,  note  the  distinction  between  "  exactly  di- 
visible "  applied  to  algebraic  forms,  and  stibmu/fiple  applied 
to  numbers.  It  is  not  even  true  that  the  highest  common 
submultiple  of  two  niimbers  which  are  obtained  from  the 
substitution  of  particular  numbers  for  the  numerical  sym- 
bols in  two  analytical  functions,  is  the  same  number  that 
would  be  obtained  by  substituting  the  same  values  in  the 
highest  common  factor  of  the  two  algebraic  forms ;  nor 
would  it  be  possible  to  make  a  definition  of  the  algebraical 
highest  common  factor,  so  that  this  should  be  true. 

The  investigations  immediately  following  apply  only  to 

integral  functions. 

A 

252.  If  A  and  1>  be  integral  functions  of  x,  and  --  =  Q, 


ALGEBRAIC   DIVISION.  155 

()  is  a  stirpal  but  not  necessarily  an  integral  function  of  x. 
(  Vide  §  169.) 

When  Q  is  an  integral  function  of  the  variables,  A  is 
said  to  be  exactly*  divlsihle  by  D. 

When  ^  (x)  cannot  be  transformed  into  an  integral  func- 
tion, it  is  said  to  be  essentially  fractional,  or  fractional. 

An  essentially  integral   function   cannot   be   identically 

{vide  §   40)    equal   to  an  essentially  fractional  function. 

A 
In  —  =  (),  if  all  the  functions  are  integral,  the  degree  of 

Q  is  the  degree  of  A  minus  the  degree  of  D. 

If  the  degree  of  yl  ?  9  less  than  the  degree  of  D,  Q  is 
essentially  fractional. 

253.  If  ^  =  PD  +  P  (all  integral  functions)  P  is  ex- 
actly divisible  by  P  or  not,  according  as  A  is  exactly 
divisible  l)y  P  or  not. 

For,  since  A  =  pp  _]_  p 

A_ 
P 


PP 

+ 
P 

R 

=  P 

+ 

R 
P 

A 

P  -- 

P 

, 

P 

P 

A  ■    ■  R  .    . 

therefore,  as  — •  is  integral  or  not,  —  is  integral  or  not. 
'         i>  °  'Z>  " 

254.    Fundamental  theorem  in  algebraic  division  :  — 

~  =  Pm-n  +  -^  ,  where  m  >  n, 

and  where  R  vanishes  or  is  an  integral  function  of  degree 
<  n. 


*  There  are  not  the  same  ohjections  to  this  phrase,  as  against  terming 
one  numher  exact  and  another  inexact.     Of-  Sections  1  and  80. 


156  NUMBER    AND   ITS    ALGEBRA. 

(The  subscripts  represent  the  respective  degrees  of  the 
functions.) 

Arranging  A,„  and  X*,,  according  to  descending  powers  of 
the  variable,  we  would  get  by  dividing  the  first  term  of  A^ 
by  the  first  term  of  i>„ , 

■^m  —  1^ -^  -'^H  ~r   -"'TO  — 1  (at  utmost). 

Dividing  by  D„  gives 

/f  7? 

B   ~     '"-"^    J)     • 

Moreover,  this  result  can  occur  in  only  one  way ;   for,  if 

—  =  PH — —^=F'A — ^,   where   the    functions    satisfy  the 

D  1)  D  .  ■ 

foregoing  conditions,  then  Avould 

-D      r,,      R'       -K  /i           1  i-      ^-         T>/  ,    -K    from    each\ 
P—P'= by    subtracting   P  -| ■  ; 

D        D  \  D      member    j 

-pt r> 

and  therefore  P~P'^=  — .,  which  is  impossible;  since 

7?'       7? 

P—P'  is  an  integral  function,  and  cannot  be  inte- 

D 

gral,  since  the  degrees  of  P'  and  R  are  less  than  the  degree 

of  D. 

1  P 

255.  If  —  =  P-\ — - ,  the  degrees  and  character  of  the 

functions  being  as  stated  in  the  preceding  section,  P  is 
called  the  integral  quotient  and  R  the  remainder  (^par 
excellence). 

Plainly,  the  necessary  and  sufficient  condition  for  "  ex- 
act divisibility  "  is  that  the  remainder  vanish. 

256.  Example  of  the  "  long  rule "  for  division  of  inte- 
gral functions  :  — • 

Divide    \  x^  -f  ^\  j-ir  +  ^\  /f  ^'J  i  ^  +  i  2/- 


ALGEBRAIC    DIVISION.  157 

The  work  may  conveniently  be  arranged  thus  :  — 


1 

4 

1 
4 

-X^lJ 

+ 
+ 

1       i/3 

1.2   U 

-I 

xhj  — 

-  tV  y 

\^y 

1      ,,3 

T^  y 

^  a;2  —  1  a-y  +  1  y2 


0    +    0,         =  7? ; 

therefore  the  latter  function  is  an  exact  divisor  of  the 
former. 

257.  The  special  case  of  the  division  of  the  general  inte- 
gral function  of  the  ?«th  degree  by  a  binomial  divisor  of  the 
1st  degree,  of  form  x  —  a,  is  of  extreme  importance. 

If  the  student  will  closely  examine  his  results  in  the 
operation  {ax''  -\- hx''-~'^-\-  ex"--  -\-  .  .  .  Ix  +  A}  ^  (x  —  a), 
he  Avill  discover  the  following  general  laws  :  — 

The  degrees  of  the  terms  of  the  integral  quotient  regu- 
larly descend. 

The  first  coefficient  of  the  integral  quotient  is  the  first 
coefficient  of  the  dividend. 

Each  subsequent  coefficient  is  the  next  preceding  multi- 
plied by  a,  -{-  the  corresponding  (in  orxler,  not  degree,  of 
term)  coefficient  in  the  dividend. 

The  remainder,  if  it  does  not  vanish,  may  be  obtained 
precisely  as  if  it  were  a  subsequent  coefficient. 

Care  must  be  taken  to  supply  by  zeros  any  lacking  terms 
in  a  particular  case. 

Example.  —  Divide  3  a;^  +  5  x^  —  9  x  -\-  11  by  x  —  2. 

-|-3-|-0-|-       5    —       9-|-ll...     (Coefficients  of  dividend). 

I      Ci    _i      -|  <>    _|_    O  <     _[_    KA  (Each  preceding  number  in  third  line 

~^         ~r       '^      r  ~r  ...  multiplied  by  2). 


-1-3  +  6+17+25+61 


158  NUMBER    AND    ITS    ALGEBRA. 

Therefore  integral  quotient  =  S  x^  -\-  6  x^  -}-  17  x  -\-  25, 
and  remainder  =  -f  Gl. 

258.  The  general  process  which  disj)layed  the  foregoing 
theorem  proves  *  also  the  following  :  — 

Remainder  Theorem.  —  If  any  integral  function  of  x 
be  divided  by  x  —  a,  the  remainder  is  the  same  function  of 
a  as  the  dividend  is  of  x.  That  is  to  say,  the  remainder 
may  be  obtained  by  substituting  a  for  x  in  the  dividend. 

Thus,  in  the  example  above, 

61  =  3  (2y  +  5  (2)2  _  9  (2)  +  11. 

If  the  divisor  were  x  -\-  2,  we  need  only  consider  x  -\-  2 
=  X  —  (—  2),  where  a  is  —  2. 

For  instance,  {3x* -\-  ox^  —  9  x -\- 11}  -^  (.y  +  2) 

gives  +3+0+5-9  +  11 

_  6  +  12  -  34  +  86 
_|_  3  _  6  +  17  -  43  +  97 

Therefore  integral  quotient  =  3  x^  —  6  ./■-  +  17  ic  —  43, 
and  remainder  =  +  97. 

And,  in  accordance  with  the  remainder  theorem, 

97  =  3  (-  2)-'  +  5  (-  2)2  -  9  (-  2)  +  11. 

259.  The  remainder  theorem  is  clearly  proved  in  the 
process  of  dividing  the  general  function  of  x  of  the  ?ith 
degree  by  x  —  a;  but  on  account  of  its  fundamental  im- 
portance in  the  theory  of  equations,  I  transcribe  an  inde- 
pendent proof  :  — 

Let  <f>,^  (x)  be  an  integral  function  of  x  of  the  nth 
degree ;  then 

*  Proved  iudependently  in  Section  359. 


KEMAINDER   THEOllEM.  159 

^"  ^'  '  =  Qn-\-\ where  R  does  not  involve  x ; 

X  —  a  X  —  a 

therefore,  (/>„  (.r)  =  <?«  _  i  (a?  —  «)  +  ^• 

But  this  equation,  being  an  identity,  holds  Avhen  cc  =  a, 
when,  (/)„  («)  =  0  +  -^j 

which,  remembering  the  meaning  of  <^„  (o),  is  merely  an 
algebraic  statement  of  the  ''remainder  theorem." 

The  full  meaning  of  this  statement  must  not  be  missed ; 
for  it  at  once  declares  the  remainder  when  </>  (x)  -^  (x  —  a), 
and  the  value  of  ^  (x)  when  x  —  a:  —  the  statement  is 
<^  {(I)  =  E. 

Thus  in  the  preceding  examples 

3  ^^  +  5  U--  —  9  a;  +  11  =  Gl  when  x  =  1,  and  =  97  when  x 
_  _  2. 

This  method  of  calculating  the  value  of  an  integral 
function  of  x  for  a  particular  value  of  the  variable  gen- 
erally saves  work  in  comparison  with  direct  substitution. 

260.  Prove :  If  an  integral  function  of  x,  ^  (x),  be  di- 
vided by  aa;  -f  J,  /      /  \ 

261.  Note  that  if  <^  (x)  vanishes  for  any  value  of  a-, 
say  V,  then  upon  division  by  a;  —  r,  7?  =  0,  and  inversely. 

262.  If  «!,  ^2,  «3  .  .  .  a^  be  r  different  values  of  x,  for 
Avhich  an  integral  function  of  x  of  the  ?ith  degree  vanishes 
where  n  >  r,  then 

(|)  (.7-)  =  {x  —  «,)  {x  —  Qa)   ...    {x.  —  a;)f„  _  r  (x), 
where/,,  _^  (x)  is  an  integral  function  of  x  of  the  (n  —  r)th. 
degree.     And  when  71  =  r, 

^  (x)  =  (x  —  «i)  (x  —  a^)   .  .   .   (x  —  a„)  /,  (;x), 
where  /,  (x)  must  be  a  constant.     (But  see  §  268.) 


160  NUMBER    AND    ITS    ALGEBRA. 

263.  An  integral  function  of  any  number  of  variables 
is  called,  homogeneous  when  the  degree  of  every  term  is 
the  same  ;  e.g.,  ax  -\-  hi/,  or  ax^  -\-  hxy  -(-  y^. 

264.  Prove :  I-f  each  variable  in  a  homogeneous  function 
of  the  nth  degree  be  multiplied  by  iii,  the  result  is  the 
same  as  if  the  function  were  multiplied  by  ?>i". 

Also :  The  product  of  two  homogeneous  functions  of  the 
mt\\  and  nth.  degrees  respectively,  is  a  homogeneous  func- 
tion of  the  (jn  -\-  ?^)th  degree. 

Let  this  last  theorem  always  be  applied  to  test  the  accu- 
racy of  distribution  of  a  jjroduct  of  homogeneous  functions. 

265.  An  integral  function  is  called  symmetrical  with 
respect  to  its  variables  when  their  interchange  leaves  the 
function  unaltered.  Several  approximations  to  symmetry 
have  received  special  names ;  e.g.,  if  a  function  be  not 
altered  except  in  sign  by  interchange  of  variables,  it  is 
called  alternating.  Functions  are  often  both  homogeneous 
and  symmetrical. 

266.  From  the  definition,  it  follows  that  the  sum,  differ- 
ence, product,  or  quotient  of  two  symmetrical  functions  is 
a  symmetrical  function,  —  a  useful  rule  in  testing  and 
abbreviating  algebraic  work. 

Since  symmetry  concerns  only  coefficients,  general  forms 
are  easily  written  down. 

Write  down  the  general  integral  symmetrical  function  of 
X,  y,  z  of  third  degree. 

267.  Since  the  coefficients  are  independent  of  the  varia- 
bles, if  two  integral  functions  are  equal  as  an  identity  (vide 
§  40),  and  the  coefficients  of  one  are  determined  by  any 
means,  then  these  coefficients  are  determined  once  for  all. 

This  theorem  has  been  called  (not  very  happily)  the 
Theorem  of  Undetermined  Coefficients. 


UNDETERMINED   COEFFICIENTS.  161 

It  is  most  useful  even  in  its  elementary  applications  to 
integral  functions,  and  becomes  an  indispensable  instru- 
ment in  dealing  Avith  infinite  series.* 

For  example  :  Required  the  product 

(^  +  2/  +  -)  (^''  +  y'  +  -'  -  ^y  -  ^-  -  y^) ; 

we  can  write  down  by  symmetry 

(x  +  y  +  z)  (x^  -\.  ,f  ^  z^  -  xy  -  xz  -  yz)  =  A  {x^  +  7/ 

+  ^3)  +  ^  (xhj  +  x-'z  +  xy^  +  xz''  +  y-'z  +  zhj) 

-\-  Cxyz. 
Since  this  identity  must  hold  for  all  values  of  x,  y,  z, 

taking 

X  =  1,    y  =  0,    z  =  0,    gives    1  =  A. 

Putting  x  =  1,  y  =  1,  and  ;v  =  0,  and  using  the  discov- 
ered permanent  value  of  A,  we  have 

*  Since  the  whole  matter  of  infinite  series  is  postponed  to  subsequent 
studies,  this  subject  cannot  be  entered  upon  further  than  to  caution  tlie 
student  that  in  sucli  an  algebraic  statement  as 

=^  \ -\-  X -\- X- -\-  x^  +  x^  .  .  .  it  is  never  to  be  understood  that 

1  —  X  1  —  X 

equals,  or  even  approximates,  the  infinite  series  unless  the  series  be  con- 
vergent; i.e.,  unless  the  sum  continually  approximates  a  definite  limit. 
Evidently  if  a;  >  1,  it  would  be  absurd  to  take  the  above  statement  into 
consideration  for  a  moment.  In  fine,  such  statements  are  understood  as 
plainly  concerning  only  such  values  of  the  variable  as  make  the  series 
convergent.  Compare  various  obvious  ellipses  common  in  all  expression 
of  thought. 

Let  this  be  the  student's  reply  to  the  cavilling  he  may  sometimes  hear 
upon  this  matter. 

Of  course  if  the  remainder  is  added  at  any  point,  the  expression  is  an 

identity,  always  true ;  e.g., =  1  +  x  +  x'^  -\ — ;  thus,  if  x  =  10, 

1  —  X  1  —  X 

we  have =  1-1-10-1-  100  -{-  1000  +  more  and  more  untrue,  the  more 

numerous  the  terms;  but  if  the  remainder  be  added  at  any  stage,  we 
have  a  true  equation: 

_L^  1  +  10  + 100+ T<^^^  +  ffl  +  ^o_lM  =  _  1. 


162  NUMBER    AND    ITS    ALGEBRA. 

1  ^1  +  1;  (^i  +  1  -  1;  =  1  (1  +  1;  +  i>Hi  +  1; ; 

or  2  =  2  +  2i?5 

therefore  i?  =  0. 

Using  these  determined  values  of  A  and  B  and  x  =  1, 
9/  =  1,  z  =  1,  we  get, 

(1  +  1  +  1)0  =  1(1  +  1  +  1)+  C; 
therefore  C  =  —  3. 

Therefore  the  required  product  is  a;^  +  y^  +  z^  —  3  xj/z. 

268.  Returning  now  to  Section  262,  it  is  plain  from  Sec- 
tion 267  that/o  (x)  must  equal  the  coefficient  of  x"  in  <^  (x). 

The  "  if "  in  Section  263  must  be  carefuPy  noted.  It 
has  not  been  shown  that  n  integral,  1st  degree  functions 
can  be  found,  such  that 

4>n  {x)  =  A-  {x  —  «i)  {x  —  «o)  (.r  —  03)   .   .   .   (x.  —  a,). 

This  question  is  also  deferred  to  subsequent  studies  in 
Theory  of  Equations,  when  it  will  be  proved  that  every 
equation  has  a  root,  and  that  every  equation  of  the  ni\\ 
degree  has  n  roots  (all  of  Avhich  need  not  be  different). 

By  a  root  of  the  equation  ^  (x)  =  0  is  meant  a  value  of 
the  variable  which  causes  the  function  to  vanish ;  that  is, 
satisfies  the  equation  ^  (x)  =  0.  AVe  have  seen  (§  261), 
that  when  an  integral  function  of  x  is  exactly  divisible  by 
x  —  a,  a  is,  2^,  root  of  the  equation,  and  inversely. 

The  general  formal  proof  that  "  every  eqiiation  has  a 
root  "  must  be  postponed ;  yet  Ave  might  almost  assume 
the  fact  as  implicit  in  the  Principle  of  Continuity  (§  103). 
Assuming  this,  we  can  prove  that  every  integral  equation 
of  the  «th  degree  has  n  roots,  and  no  more. 

Let  a  be  one  root ;  then, 

,/;  (.r)  =  (x-  —  «)/„_i(.r); 


ROOTS    OF    INTEGRAL   EQUATIONS.  163 

but/„_i(.r)  must  have  a  root,  and  so  on  for  u  roots  (some 
of  which  might  be  repeated),  and  a  constant  factor,  fg  (x) 
(vide  §  262).  Moreover,  ^„  (ic)  cannot  have  more  than  7i  dif- 
ferent roots,  because  if  any  integral  function  of  ?/th  degree 
vanish,  for  more  than  ii  values  of  the  variable,  it  must 
vanish  identically;  that  is,  for  all  values  of  x  (i.e.,  every 
coefficient  in  form  cr"  -j-  ^*"~'  +  ex""-  -\-  .  .  .  -\-  dx  -{-  k 
must  be  zero).     For,  let 

<|)„  (a:)  =  a  (x  —  i\)  (x  —  r.)  {x  —  r^)  .  .  .  (x  —  ?'„)  ...  (1). 

Now,  if  possible,  let  ;•  be  another  value  of  the  variable 
for  which  the  function  vanishes.  Since  (1)  holds  for  all 
values  of  x,  then 

^  (.'')  =  f*  ('■  —  ''i)  (''  —  ^'2)  ('■  —  ''3)   ..•('•  —  r„)  =  0 ; 

and  since  each  ''  r"  by  hypothesis  is  different,  a  must  be 
zero. 

But  a  is  the  coefficient  of  the  .r"  term  in  ^„  (x).  In  this 
way,  step  by  step,  each  coefficient  in  (f),,  (x)  is  shown  to 
vanish  if  more  than  n  values  of  the  variable  satisfy  the 
equation  (|)„  (./;)  =  0. 

For  example,  x'^  —  (x -\- 1)  (x  —  1)  —  1  is  of  the  2d  degree, 
yet  plainly  it  vanishes  for  0,  1,  2,  —  and  therefore  for  all 
values  of  x. 

269.  The  preceding  section  affords  an  independent  proof 
of  the  theorem  of  undetermined  coefficients,  Avhich  may  be 
re-stated  as  follows  :  — 

Any  function  of  x  is  transformable  into  an  integral  func- 
tion in  only  one  way.  For,  if  possible,  suppose  the  two 
following  different  integral  functions,  derived  from  the 
same  function,  as  identities,  and  therefore  equal  for  all 
values  of  x  : 


164  NUMBER    AND    ITS    ALGEBRA. 

No  generality  is  lost  in  regarding  them  as  of  the  same  de- 
gree ;  for,  if  not,  it  would  simply  mean  that  the  coefficients 
concerned  were  zero.  Subtracting  the  right-hand  member 
froin  each,  we  get 

{a  —  ai)  cc"  +  {h  —  h^)  a;"-'  +  (c  —  c^)  a;"-^  -|-    .   .   . 

for  more  than  n  values  of  x. 

Therefore,  a  —  a^  =  0,  b  —  h^  =  0,  .  .  .  k  —  k^  =  0  ; 
that  is  to  say,    a  =  a^,    b  =  l/^,  .  .   .   k  =  k^. 

270.  Professor  Chrystal  remarks  at  this  point  in  his 
Text  Book  of  Algebra,  "  the  danger  with  the  theory  we 
have  just  been  expounding  is  not  so  much  that  the  student 
may  refuse  his  assent  to  the  demonstration  given,  as  that 
he  may  fail  to  apprehend  fully  the  scope  and  generality  of 
the  conclusions."  Their  utility  cannot  fail  to  be  more  and 
more  highly  appreciated  by  the  attentive  student. 

271.  (1)  Determine  the  value  of  Z;  such  that  2  a'^  —  8 
x"^  -{-  1 X  -\-  k  shall  be  exactly  divisible  by  x  -{-2.  By 
Section  259,  the  remainder  to  division  by  x  —  (—  -)  is 
2  (_  2)3  _  8  (-  2)2  +  7  (-  2)  +  /.•  =  -  62  +  k. 

If  the  function  is  to  be  exactly  divisible  by  a-  -J-  2,  this 
remainder  must  vanish,  or  —  G2  -|-  ^^  must  be  zero ;  i.e., 
k  =  62. 

(2)  In  like  manner  the  question  of  exact  divisibility 
may  be  readily  tested  : 

/y»n   j,n 

AVhen ■—  ,  72  =  y"  —  ?/"  =  0 ;  the  division  is  always 

x  —  y 

exact. 


GIVEN   ROOTS,    TO   FORM   EQUATION.  165 

When  -  ~^^^i  ,  i?  =  (_  y)"  —  ?/"  —  0,  if  n  be  even,  = 
^'  +  y 
—  2  y  if  n  be  odd. 

When  — J^^  ,    ^  =  _j/»  -}-?/"  =  2  ^"  ;     the    division    is 

■^  -  y 

never  exact. 

When  ^^li.^",  R  =  (—  yY  -]-  y''  =  0,  if  ?i  be  odd,  = 
2  ?/"  if  n  be  even. 

(3)  If  A  -~  D  gives  remainder  R,  and  B  -^  D  remainder 
i2',  show  that  AB  -=-  2)  and  RR'  -^  i>*  give  identical  re- 
mainders. 

(4)  Observe  that,  in  the  proposition  that  an  equation  of 
the  ?ith  degree  has  n  roots  and  no  more,  we  prove  that 
any  finite  number  has  n  nth  roots  and  no  more,  — ■  all  of 
which  need  not  be  different. 

To  find  these  roots  of  any  number,  a  requires  the  solu- 
tion of  the  equation  »•"  =  a,  or  ic"  —  a  =  0 ;  that  is  to  say, 
the  factorization  of  a-"  —  a  in  the  form, 

{x  —  ri)  (a-  —  7\)  {x  —  I's)  .  .  .   (x  —  ?•„). 

(5)  We  are  also  enabled  to  make  an  integral  equation  of 
given  roots.  Thus,  to  form  an  equation  whose  roots  are  0, 
+  1,  —  V2,  —  1,  we  have  simply  to  write, 

Cx  (x  —  l)(x-\-  V2)  (x  ~\-l)  =  0, 

where   C  is  any  constant    we   please;  e.g.,  thts   equation, 
taking  C  =  1,  is 

a-"  +  V2  a;3  _  a;2  _  ^2  a-  =  0 : 
or,  taking  C  =  V2, 

V2  a;-*  +  2  a;=5  ^  V2  a;-  -  2  x  =  0. 


166  NUMBER   AND   ITS   ALGEBRA. 

272.  Having  thoroughly  explained  the  meaning  of  "  ex- 
act" divisibility  as  api^lied  to  the  division  of  one  integral 
function  by  another,  the  sense  in  which  one  function 
is  termed  the  highest  common  factor  of  two  others  is 
apparent : — 

The  integral  function  of  x  of  highest  degree  which 
''exactly  divides"  each  of  two  or  more  integral  functions 
of  X,  is  their  highest  common  factor  (h.  c.  f.).  (But  see 
§  251.) 

If  the  given  functions  are  easily  resolvable  into  factors 
which  are  integral  functions  of  the  first  degree,  the  h.  c.  f. 
is  readily  taken  by  inspection;  since  it  is  simply  the 
product  of  such  of  these  first  degree  factors  as  are  com- 
mon, each  raised  to  the  lowest  power  in  which  it  occurs 
in  either  of  the  given  functions. 

Otherwise  we  may  proceed  very  much  as  in  Section  240, 
since  if  A  =  BQ-]-  R,  the  h.  c.  f.  of  A  and  B  is  the  h.  c.  f. 
of  B  and  R  :  proved  by  considering  Section  253. 

Consequently,  to  find  the  h.  c,  f.  of  two  integral  func- 
tions of  X,  A  and  B,  where  the  degree  of  B  is  less  than 
that  of  A,  we  may 

divide  A   hy  B   so  that  A  =  BQ^-^-  Ry 

and  divide  B   by  ^i  so  that  B  =  R^Q^ -\-  Rr,, 
and  divide  R^  by  R^  so  that  R^  =  R^Q^  -\-  R.,  etc.,  until 

Rn-i/Rn  gives  R„_i  =  R„  ^„+i  +  R, 
where  R  vanishes,  or  is  of  zero  degree,  that  is,  a  constant. 
In  the  latter  case,  there  is  no  h.  c.  f. ;  in  the  former  R^  is 
the  h.c.f.  For  by  Section  253,  A  and  B,  B  and  ^i, 
^1  and  R2,  .  .  .  R„_i  and  R„,  are  of  descending  degree, 
and  all  have  the  same  h.  c.  f.,  and  no  factor  of  higher 
degree  than  R^  can  exactly  divide  R„.  In  case  R  is  a 
constant,  R„_i  and  R„  have  no  common  exact  divisor  other 


HIGHEST    COMMON   FACTOR.  167 

than  R ;  that  is  to  say,  there  is  no  common  clivisoi-  in  the 
sense  intended,  although  any  constant  will  ''exactly  di- 
vide "  any  integral  function  in  the  sense  of  giving  an 
integral  quotient ;  i.e.,  remainder  zero.  ( Vide  §  §  255, 
256.) 

It  follows  from  the  nature  of  this  process  of  finding  the 
h.  c.  f.  that  at  any  stage  either  divisor  or  dividend  may 
be  multiplied,  or  divided  by  any  integral  function  of  the 
variables  (of  course  including  any  constant),  provided  it  is 
certain  that  the  factor  so  introduced  or  removed  has  no 
factor  in  common  with  the  other  functions.  Any  function 
which  is  obviously  a  common  factor  of  both  dividend  and 
divisor  at  any  stage  may  be  removed  from  each,  provided 
we  multiply  the  h.  c.  f.  afterwards  resulting  by  the  re- 
moved common  factor.  In  dealing  with  factors  which  are 
constants,  regard  "factor"  in  the  sense  of  common  suh- 
multlple  of  the  coefficients.  Finally,  it  must  be  observed 
that  the  recurring  operations  are,  on  account  of  such  modi- 
fications as  have  been  ascribed,  not  divisions  in  the  ordi- 
nary sense ;  for  the  "  division "  may,  if  convenient,  be 
arrested  at  any  stage  (while  the  remainder  is  yet  of  higher 
degree  than  the  divisor),  to  remove  common,  or  introduce 
independent,  factors. 

273.  (1)  Find  h.  c.  f.  of  9  .^^  -  30  .r*  +  4.j  .7-^+  24  x  and 
15  x^  -  30  X*  —  90  a;3  +  60  x^  -f  195  a;  +  90.  (Problem 
worked  out  on  page  168.) 

Of  the  originally  removed  factors,  3  x  and  15,  3  is  com- 
mon ;  therefore,  cc^  —  3a:;^-|-3a;-|-l  must  be  multiplied  by 
3  to  obtain  the  h.  c.  f.,  3  a-^  +  9  a;^  -f  9  a;  +  3. 


168 


NUMBER   AND   ITS   ALGEBRA. 


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ALGEBRAIC    PKIMENESS. 


169 


(2)  What  is  the  necessary  relation  among  their  coeffi- 
cients in  order  that  aar-  +  hx  +  c  and  cx^  +  te  +  a  may 
have  an  exact  common  divisor  of  the  first  degree  ? 


ax 


/^_.^)' 


ax"^  -\-  hx  -|-  c 


ax 


2  I  «(«  +  g) 


X 


^^_a{a+c)\ 


cx'^-\-l>x-{-a 

cx^-\ x-\ — 

a         a 

(dividing  by 
coef .  of  X  gives) 

,    fl.  +  C 


V' 


c  a 


c  —  a  —  c-\ ^ — ! — -^Jt. 

Now,  if  the  functions  have  an  exact  common  divisor  of 
the  first  degree,  It  must  vanish ;   therefore  the  condition 

c  —  a  —  c  A ^^ ■ — ^—  =  U. 

^  b'' 

Whence  -  ab""  +  a  {a  ^  cf  =  0  ; 

or,  dividing  by  «,  {a  -\-  cy  =  b- ; 

or  ■  a  -{-  c  =  ^  b. 

274.  From  Sections  253  and  272  it  is  plain  that  the 
h.  c.  f .  of  three  integral  functions  is  the  h.  c.  f.  of  the  h.  c.  f. 
of  two  and  the  third,  and  so  on. 

275.  Integral  functions  which  have  no  common  exact 
divisor  are  said  to  be  algebraically  prime.  Many  condi- 
tions of  algebraic  primeness,   more   or  less  analogous  to 


170  NUMBER    AND    ITS    ALGEBRA. 

those  estaBlished  concerning  absolute  numerical  primeness, 
might  be  investigated. 

276.  The  precise  meaning  of  algebraic  lowest  common  mul- 
tijjle  of  two  integral  functions  will  now  easily  be  understood 
as  the  integral  function  of  lowest  degree  exactly  divisible 
by  both. 

Let  A  =  HQ  and  B  =  HQ',  where  the  symbols  repre- 
sent integral  functions,  and  H  the  h.  c.  f.  of  A  and  7>.  Let 
M  be  any  common  multiple  of  A  and  B ;  then  — 

M  =  AE,  where  E  is  an  integral  function  of  x. 

Therefore  M  =  HQE. 

But  Jf  is  an  algebraic  multiple  of  i?  =  HQ'\ 

^1        r         M        HQE       QE       ,         QE  ,   , 

therefore  ■ =  — - —  =  -^ — ,  where  -i —  must   be  an   m- 

HQ'        HQ'         Q'  Q' 

tegral  function.  But  since  Q  and  Q'  are  by  hypothesis 
algebraically  prime,  E jQ'  =  X  or  E  =  (i'X,  where  X 
is  integraL     Consequently 

Jf  =  HQE  =  HQQ'X. 

But  this  last  algebraic  statement   (translated)  declares 

that  any  common  multiple  of  A  and  B  is  the  product  of 

H,  Q,  and^',  as  defined,  and  some  other  integral  function, 

X.     Hence,  31  is  of  the  lowest  possible  degree  when  X  is 

of  zero  degree  ;  that  is,  a  constant.     And  since  constants 

are  not  altogether  ignored  in  the  desired  result,  M  is  the 

''lowest  common  multiple"  when  X=l;  that  is  to  say, — 

A  P 
since  HQQ'  =  ■ — -,  the  1.  c.  m.  of  two  integral  functions  is 

the  quotient  of  their  product  divided  by  their  h.  c.  f. 

Alternative  rules  are  similar  to  those  for  single  numbers 
{vide  §  242).  The  algebraic  1.  c.  m.  has  neither  the  practi- 
cal nor  the  theoretical  importance  of  the  algebraic  h.  c.  f. 


SCALE   NOTATION.  171 


277.  The  fundamental  theorem  in  the  expression  of 
numbers,  in  a  notation  such  as  our  common  system,  is  tlie 
following  :  — 

Any  primary  number  may  be  expressed  finitely,  and  in 
only  one  way  in  the  form  ■ — • 

^o  +  ^1  (''i)  +  ^2  OVo)  +  ^3  (rir^rs)  -j-  .  .  .  c„  ()\r.r3  .  .  .  r„), 

where  Vi,  Vo,  r^,  .  .  .  r„  is  a  series  of  primary  numbers, 
unrestricted  except  that  there  are  as  many  as  may  be 
required,  and  Co<  rj,  c^-C  r.2,  c.2<  r^,  etc. 

For  if  /  be  any  primary  number,  dividing  I  liy  r^  gives 

(1)  /    =  Co  +  Qi)\    where    CoKn; 
and  dividing  Q^  by  r^  gives 

(2)  '   Qi  =  r^  -{-  Q.,r-2   where  <?i<  r^; 
and  dividing  Q.^  by  rg  gives 

(3)  Q.  =  C.2  +  (^3/-3   where   Co<  rg; 
and  so  on  until  Qn<  r„^i  is  reached. 

(1)  and  (2)  give 

^  =-  <'o  +  (''i  +  Q'^ro)  i\  =  ^0  +  ^I'-i  +  Q-ir^ro ; 
and  svibstituting  for  Q.2  from  (3) 

1=  Co-\-  c^r^  +  c,  (ri7\)  +  Qsnr.Vs ; 

and  so  on  until  Q„_i  =  <"„_i  +  (?„'"«>  where,  writing  6'„  for 
Q„,  we  have 

I  =  Co  -\-  c^  (ri)  +  C2  (i\r.2)  +  (-3  (t\r^rs)  +   .   .  .   r„ 
(7\r^r3  .  .   .   ?•„). 

Moreover,  this  expression  is  unique  for  the  same  series 
of  r's,  because,  if  not,  let 

^0  +  <-i'-i  +  ^2  QV2)  +    .   .   .    =  Co'  +  r^'r^  +  e/ (}\ro)  +   .   .   . 


172  NUMBER   AND    ITS   ALGEBRA. 


dividing  each  member  by  ?*i,  gives 


-0 


+   q  +   f2»-2  +...=  —  +    Cl'  +    C2''-2     + 


y-l 


But  since  Cq  <ri,  and  Cq'  <  i\,  by  Section  245  the  essen- 
tially integral  and  essentially  fractional  parts  of  these 
numbers  must  be  equal  separately;  that  is  to  say, — 

£2  =  £2. ,  or  Co  =  Co  ,   and  so  forth. 

Example.  —  Express  the  number  represented  in  our 
notation  by  200  on  the  scale,  7,  3,  5,  2,  etc. 

7 1 200 
3 1 28  ...  4 
5\j)_.  .  .  1 
1  ...  4 

since  the  quotient  1  is  less  than  i\  =  2,  the  process  ter- 
minates, and 

200  =  4  +  1  X  7  +  4  (7  X  3)  +  1  (7  X  3  X  5). 

Express  100  in  the  scale  3,  4,  5,  6,  7,  etc. 

278.  A  corresponding  theorem  for  the  expression  of 
numbers  essentially  fractional  (§  245)  where  numerator  < 
denominator  (''proper"  fractions),  may  be  proved;  that  is 
to  say  :  — 

—  = r  ~  "T" r    •    •    •  '     r»'  ' 

D        ri        i\i\       r^r^n  n^'2^3   •  •  •  »«       ^ 

where  Jj' =  {riTor^  .  .  .  r„)  D, 

where  d^  <  i\  ,  d^  <  i\ ,  etc.,  and  where  /  may  vanish.  The 
general  proof,  and  demonstration  that  /  =  0,  when  r,  r^  is 
...  r„  is  a  multiple  of  B,  is  left  as  an  exercise  for  the 
student. 


RADIX   NOTATION.  173 

Example.  —  In  this  way  express  7/  10  in  scale  5,  7,  9, 
11,  13,  15,  17,  .  .  . 


5 

035 

3 

.  .  .  5 

7 

10)35 

3 

.  .  .  5 

9 

10)45 
4 

.  .  .  5 
11 

10)55 

5 

.  .  5 

evidently  this  cannot  terminate,  because  j\,  Vo,  r^  .  .  .  r„ 
can  never  be  a  multiple  of  10. 

Therefore,    7/ 10  =  |  +  ^-i.  +  ^^  +  ^rj:^^,  • 

279.  Since  the  scale  of  r's  may  be  arbitrary,  it  may  be 
chosen  so  that  all  the  fZ's  shall  be  one.  Decompose  7/10 
as  a  sum  of  fractions  with  unit  numerators. 

280.  When  the  scale  of  notation  is  constant ;  that  is, 
when  all  the  r's  are  equal  is  the  important  special  case  of 
the  foregoing  Theorem. 

In  this  case 


7=  c^r'^  +  c^r  +  Coj"  +  c^r^  +    •   •   •   cj 


1 


and  —  =  d^r-'  +  cLr-'-  +  d,r-^  +   •   •  •   dj--"  +    -^ 


Avhere  /  may  be  zero. 

Here,  of  course,  is  recognized  our  system  of   notation, 
where  r  =  10. 


174  NUMBER   AND    ITS   ALGEBRA. 

It  is  our  custom  to  omit  the  7''s,  whose  powers  are  under- 
stood from  the  order  of  the  c's  and  c^'s,  the  proper  place 
being  displayed  by  never  failing  to  express  the  zero  when 
any  c  or  d  has  this  value.  Also  we  omit  the  signs  of  addi- 
tion, and  write  the  integral  series  from  right  to  left,  so 
that  it  may  be  regularly  continued  by  the  fractional  series, 
a  mere  point  amply  serving  to  separate  the  two.  In  this 
way  the  powers  of  the  radix  or  base  decrease  by  ones  from 
left  to  right,  thus  : 

Cn  ''"  +  ^„-i ''""'  +  •  •  •  ^2  '•''  +  ^1  ^^  +  -^0  '>'"  +  ^1  ^~^  +  ^2  r-^ 
-\-  d^r-^,  etc. ;  or,  omitting  -f's  and  r's  and  pointing  off  the 

281.  From  the  condition  that  the  c's  and  d's  must  all  be 
less  than  r,  it  is  obvious  that  in  any  such  notation  r  —  1 
figures  are  required  to  uniquely  designate  the  jDOssible 
values  of  c's  and  c^'s. 

It  is  also  plain  that  all  the  rules  of  the  decimal  algo- 
rithm apply  to  any  other  base,  say  12,  except  that  the 
"  carriages  "  would  go  by  12's  instead  of  lO's.  Of  course 
for  radix  12,  tAvo  new  digit  figures  would  be  required  ;  and 
for  radix  2,  symbols  for  1  and  0  only  could  be  used.  Thus 
teM  on  the  binary  scale  would  be  1010  ;  that  is, 
1  X  2^  +  0  X  2-2  +  1  X  2^  +  0  X  2°. 

282.  Example. — Express  102305  (radix  ten)  on  base 
twelve  121102305 


■19    ^KOK  K 


12  710  ...  5 


12  59  ...  2 


(using  a  and  h  as  digits  for  ten  and  eleven). 

Therefore  102305  (r  =  ten)  is  4  ^^  255  (r  =  twelve). 


RADIX    NOTATION. 


175 


Inversely,  to  express  4  h  255  (radix  twelve)  on  decimal 
base. 

Consider  the  expression  means  (using  our  common  nota- 
tion for  calculation), 

4  (12)*  +  4  (12/  +  2  (12/  +  5  (12;  +  5 ; 
or,  performing  the  indicated  operations. 


5  = 

5 

60  = 

5(12) 

288  = 

2  (12)2 

19008  = 

11  (12)3 

82944  = 

4  (12)* 

102305 

If  the  student  will  refer  to  Sections  257-259,  he  will 
notice  that  the  remainder  theorem  yields  an  easier  way 
for  this  calculation.  (1)  The  problem  is  merely  the  evalu- 
ation of  the  given  function  of  twelve.  We  may  therefore 
write :  — 


4,    11, 


5, 


48,    708,    8520,    102300, 
4,    59,    710,    8525,    102305. 

It  would  be  good  practice  to  Avork^  out  as  follows 
the  duodecimal  algorithm,  on  its  own  merits,  "carrying 
twelve  —  but  from  our  fixed  habit  of  thought  this  is  much 
more  difficult :  — 


m 


«|4/>255 
a   5  b  00  . 

.  .  5 

a   713  . 

.  .  0 

a   86  . 

a  a  . 

1  . 

.  3 

.  -J 
.  0 

that  is,  4  h  255  (radix  twelve)  is  102305  (radix  ten). 


176  NUMBER   AND    ITS    ALGEBRA. 

283.  Fractions  expressed  in  sucli  notations  as  are  under 
discussion  are  called  radix  fractions,  decimal  if  the  base  is 
ten,  duodecimal  if  the  base  is  twelve,  etc. 

A  fraction  ^^  expressed  as  a  radix  fraction  cannot  termi- 
nate unless  iVr"  is  a  multiple  of  D ;  for 

±=.d,d,d,  '      ^^ 


JJ  ^   ^   "  '   r"i> 

Multiplying  each  member  by  ?•"  reduces  the  radix  fraction 
part  of  the  right-hand  member  to  an  integer,  giving 

^-_  =  d^d„ds   .  ,   .  d,^-\-  j-  , 

where  tZj  do  .  .  .  is  the  integral  part  of  the  quotient ;  and 
there  must  be  a  fractional  part  unless  lir"  is  a  multiple 
of  D.  Also  if  N I D  is  in  "lowest  terms,"  i.e.,  if  N  be 
prime  to  D,  it  is  plain  that  a  radix  fraction  cannot  termi- 
nate unless  ?'"  is  a  multiple  of  I).  Nor  can  ?•"  be  a  multiple 
of  D  unless  it  be  resolvable  into  powers  of  primes  which 
are  jDrime  factors  of  ;•.  For  example,  to  express  N j  D  (in 
its  lowest  terms)  as  a  decimal  fraction,  we  must  have 
D  =  2''  5",  where  either  x  or  y  may  be  zero. 

N  . 
284.    If,  when  the  proper  fraction  —  in  its  lowest  terms 

is  expressed  as  a  radix  fraction,  the  latter  does  not  termi- 
nate, its  digit  figures  must  repeat  in  a  cycle  of  not  more 
than  _D  —  1  figures.  For,  evidently  only  D  —  1  different 
remainders  can  occur,  and  Avhen  one  recurs,  the  figures  of 
the  quotient  must  repeat.  Such  radix  fractions  are  called 
repeating,  recurring,  or  circulating. 

The  repeating  period  may  begin  at  once,  or  may  begin 
after  figures  which  do  not  repeat,  —  commonly  distin- 
guished  as  ^^wre   and   mixed   circulates.       The    repeating 


RADIX    FRACTIONS.  177 

period  is  sometimes  called  perfect  when  it  consists  of  the 
full  complement  {D  —  1)  of  figures.  The  repeating  period 
is  denoted  by  dotting  its  first  and  last  figures. 

This  subject  could  be  better  discussed  in  connection 
with  infinite  series,  and  "geometrical"  progression;  but 
repeating  decimals  occur  so  frequently  in  practice  that 
their  reduction  to  simple  fractions  cannot  be  left  in  the 
dark. 

Consider  — 

3[lj 1 

0-333333  +  3  X  ]^Q6  •  •  •  1/S    =0-3 

7L1: 1 

0-142857142857  +  ^-TTTTTT  •  •  •  1/7    =0-142857 


<xioi- 


1111- 


'•'''^'+1T^^  ...1/11  =  0-09 

24  [Ij 

0-04133  + ^—-  ...  1/24  =  0-0413. 

^  24  X  10^  ' 

The  remainders,  inexpressible  as  a  radix  fraction,  may 

be   introduced    at    any   point.       I  express   them   to  avoid 

discussion  of  infinitesimals;    and  if  regarded   as  implicit 

in  the  notation  of  the  repeating  decimals,  the  reasoning 

in  this  section  is   exact  in  terms   of  thought  familiar  to 

beginners. 

Now  1     =  0-111  + 

^^    =0-010101  + 

^1^   =  0-001001  + 


j__  =  0-00010001  + 


The   law   is    plain,   and    furnishes   a  way  to  transform 
repeating  decimals  into  simple  fractions. 


178  NUMBER    AND   ITS   ALGEBRA. 

For  example :  Express  0-324  as  a  common  fraction  in  its 
lowest  terms.  Evidently  0-324  =  324  x  0-OOi.  But  0-6oi 
=  ^iT ;  therefore  0-324  =  §|f  =  ^Yt  =  if- 

Hence  the  rule:  To  express  any  pure  circulating  deci- 
mal as  a  common  fraction,  write  the  repeating  period  for 
numerator,  and  for  denominator  as  many  nines  as  there  are 
decimal  places  in  the  repeating  period.    ' 

To  find  the  rule  for  mixed  circulates,  consider :  — 

F  =  0-3i8 

1000  F  =  318 'ISIS  4-  ft  ^^'^^^^6  these  remainder  frac- 

-tf\  p< S-1  S1  S  -U  V  (       tions,  /,  are  absolutely  the 

^^ — "^     '"■     1       same. 

therefore  990i^=315 

and  F=U^  =  ^^. 

Again  consider :  *  — 

i^=0-03G93is 
10,000,000  F  =  369318-18  +/  )  ^^'^  remainder  frac- 

100,000  i^=      3693-18 +  /•(     ['^'"f    t''   ^^""^ 

,,         „         — — — —^  I       lutelv  the  same. 

therefore      9900000  F  =  365625  ^ 

and  F=  r^f^-Aesji-  —  jis 

!?  9  00000    —    35  J" 

Hence  the  rule  .-  To  express  a  mixed  circulating  deci- 
mal as  a  common  fraction,  subtract  the  non-repeating  part 
from  the  whole  circulate  for  the  numerator,  and  for  the 
denominator  write    as    many  nines  as  there  are  decimal 

*  The  same  result  may  he  obtained  thus : 

0-03693i8  =  0-0.3693  +  Q-OOOOOiS 

^3fi9  3._  _| 1        V  n-i«! 

—   TOOOOO         '      10000(7       A    U   lO 
3fi93  I  1  s/18 

—  T05p(J  +  JT^h^V  —  TT  fff  f  §7  =  ^V 


REPEATING   RADIX   FRACTIONS.  179 

places  in  the  repeating  period,  followed  by  as  many  zeros 
as  there  are  places  in  the  non-repeating  part. 

285.  Inasmuch  as  we  have  seen  that  any  integer  is  ex- 
pressible in  only  one  way  in  any  radix  scale,  it  is  clear 
that  a  common  fraction  in  any  scale  of  notation  is  expres- 
sible as  a  common  fraction  in  any  other  scale.  Con- 
sequently any  terminating  or  repeating  radix  fraction  in 
any  scale  transforms  into  a  common  fraction,  and  therefore 
into  a  terminating  or  repeating  fraction  in  any  other  scale. 

Note  carefully  that  a  terminating  radix  fraction  in  one 
scale  need  not  transform  into  a  terminating  radix  fraction 
in  another  scale,  but  into  a  terminating  or  repeating  radix 
fraction. 

286.  To  transform  a  fraction  from  one  scale  to  a  radix 
fraction  in  another,  simply  multiply  by  the  new  base,  and 
the  fractional  part  of  the  product  again  by  the  new  base, 
and  so  on.  The  integral  parts  of  these  products  in  due 
order  are  the  figures  of  the  transformation.  For  example, 
to  express  |  as  a  duodecimal  fraction  :  — 

3  X  12  =  4i 
i  X  12  =  6, 

therefore  |  =  0.46  (radix  twelve). 

Or  again,  to  express  0.13  as  a  radix  fraction  in  the  seven 
scale  :  —  q  -l^g 

7 


0.91 

7 

6.37 

7 

2.59 

7 

4.13 


180  NUMBER    AND    ITS    ALGEBRA. 

here  13  recurs,  and  the  fraction  repeats  this  period,  so,  — 

0.13  (radix  ten)  =  0-6024  (radix  7). 

To  prove  the  propriety  of  this  process,  consider  a  proper 
fraction,  F,  and  let — • 

r         I-         r 

in  some  new  scale  of  notation  whose  base  is  r. 

Then  rF  =7-,  +  ^^^+   .  .  . 

r  /•- 

say  vF  =  x^  -\-  F,   where   F  must  be   a  proper  fraction ; 
therefore  x^  is  the  integral  part  of  rF. 

Again  rF  =  :r„  -[-  ^  -f  "li.  _)_ 

And  in  like  manner  Xn  is  the  integral  part  of  rF ;  and 
so  on. 

287.  If  /  be  any  integer,  and  s  the  sum  of  its  digits,  and 
r  the  radix  of  the  scale  of  notation,  then  the  remainder  of 

=  the  remainder  of  — ^^ —  . 

r - 1  r -1 

For,  let  /  =  Co  +  ("i  ?'  4-  ^2  '"^  +  ^ 3  '"^  +   •  •  •  <'n  ^■''• 
Subtracting  the  sum  of  the  digits  from  each  member  of 
the  equation  gives 

/  _  s  =  q  (;•  -  1;  +  r^  (/-'^  -  1)  +  ^3  (>■''-  1)  +  .  .  .  c,^  (/■"  -  1). 

Since,  by  Section  271,  each  term  of  the  right-hand  mem- 
ber is  a  multiple  of  (r  —  1),  if  we  divide  each  member  by 
(?'  —  1)  (or  any  submultiple)  we  get 

=  some  integer. 

r-1       r-1  " 

Therefore,  by  Section  245.  the  essentially  fractional  parts 
of  I J  r  —  1  and  s  J  r  —1  must  be  equal. 


EEMAINUEUS   TO    NINE.  181 

288.  From  this  theorem  follows  the  special  corollary 
that  in  our  decimal  notation  any  integer  and  the  sum  of 
its  digits  give  the  same  remainders  to  9  or  3. 

This  is  the  reason  of  the  familiar  rule  for  "casting  out 
9's,"  in  order  to  test  the  accuracy  of  calculations. 

If  P  =  MN  =  9  a;  +  ^>  =  (9  y  +  m)  (9  s  +  n),  where  jh 
111,  and  n  are  the  respective  remainders  to  9  of  P,  M,  and 
N,  it  follows  that  2^  ^^tl-  "^'^  gi'^e  the  same  remainders  to  9, 

since         ^  x  -\-  p  =  (9  y  +  ?»)  (9  z  -\-  n) 

=  9-1/."  +  9  (/?y  +  mz)  -J-  17171 

=  9  (9  l/Z   -(-   711/  -\-  mz)   -\-   77171. 

In  practice,  find  7?,  171,  and  n,  not  hy  dividing  P,  M,  and 
N,  by  9,  but,  in  accordance  with  the  theorem,  by  dividing 
their  digit-sums  by  9  ;  ''  cast  out "  the  nines.  It  is  plain 
also  that  the  remainder  to  9  (or  3)  of  ^  +  i>  -|-  C  equals 
the  like  remainder  to  a  -\-  h  -\-  c,  where  a,  h,  and  c  are  the 
respective  remainders  to  A,  B,  and  C. 

Therefore  to  test  addition  :  — • 

(1) 


8277 

remainder  to  9 

.     .     6 

3485 

remainder  to  9     . 

0 

•          >          ^ 

7146 

remainder  to  9     . 

.     .     0 

8036 
26944 

remainder  to  9     . 

.     .     8 
16 

The  sums,  26944,  and  16,  each,  give  the  same  remainder, 
7  ;  consequently  the  addition  *  is  probably  correct,  —  only 
probably,  because  this  check  could  not  take  note  of  an 
error  of  any  multiple  of  9,  or  compensating  errors,  or 
transposition  of  figures. 

*  Strictly,  partial  additions,  and  associations  to  suit  our  notation. 
(Fide  §§72,  73.) 


182  NUMBER    AND    ITS    ALGEBRA. 

(2)  To  test  subtraction  :  — 

87235  remainder  to  9  ...  7 
14505  remainder  to  9  ...  3 
72670  X 

The  difference,  72670,  and  4  give  same  remainder,  4,  to  9. 

(3)  To  test  multiplication :  — 

349751  remainder  to  9     .     .     .       2 

28637  remainder  to  9     .     .     .       8 

10015819387  l6 

The  product  and  16  give  the  same  remainder,  7. 

(4)  To   test   division,    let   the    student    prove    that    if 

The  remainder  to  nine  of  P  =  remainder  to  nine  of 
{qd  4-  ?•),  where  q,  d,  and  r  are  the  respective  remainders 
to  nine  of  Q,  D,  and  B.     Thus  to  test  the  division, 

27220662   ..o  ,  398 


47923  47923 

remainder  to  47923  (divisor)     =^  7 

remainder  to  568  (quotient)  =  1 

remainder  to  398  =  2 

remainder  to  7  X  1  +  2  =0 

remainder  to  dividend  =  0 

289.    Problems  :  — 

(1)  Expressed  in  a  certain  scale  seventy-nine  becomes 
142,  what  is  the  radix  ? 

(2)  In  what   scale   of   notation    does   301   express  the 
second  power  of  an  integer  ? 

(3)  Dediice    a   test  of    multiplication  by   "casting  out 
elevens." 

(4)  Prove  that  any  integer  of  four  digits  in  the  scale  of 


ALGEBRAIC    SQTTARE   ROOT.  183 

ten  is  a  multiple  of  7,  if  its  first  and  last  digits  be  equal, 
and  the  hundreds  digit  twice  the  tens  digit. 

(5)  In  ten  scale  a  number  of  6  digits  whose  1st  and  4th, 
2d  and  5th,  3d  and  6th  digits  are  respectively  the  same,  is 
a  multiple  of  7,  11,  and  13. 

290.  The  common  process  of  finding  algebraic  square 
roots,  cube  roots,  etc.,  is  familiar  to  all,  most  text-books 
making  far  too  much  of  it.  The  method  has  little  interest, 
theoretical  or  practical.*  Even  the  analogous  numerical 
calculations  are  better  dispensed  with,  if  a  table  of  log- 
arithms is  at  hand ;  and  the  method  for  the  algebraic 
problem  is  rendered  superfluous  by  the  simpler  method 
of  "undetermined  coefficients."  AVe  consider  only  cases 
where  the  function  is  a  perfect  square,  because  further 
discussion  would  take  us  into  the  question  of  infinite 
series. 

Example.  —  Eequired  the  algebraic  square  root  of  — 

^  12        3       9 

If  a  "  perfect  square,"  the  root  must  be  of  the  form,  ax^ 
-\-l>x  -\-  c,  the  square  of  which  is  (rx*  +  2  abx^  -\-(2  ac  -f  Z»^) 
a;2  _|_  2  l,e.v  +  cl  The  corresponding  coefficients  must  be 
equal ;  therefore,  o  =  1.  2  ab  =  1  .-.  h  =  1  /  2.  2  he  = 
—  1/3. '.6=— 1/3;  therefore  the  required  square  root  is 

x^-\---  1/3. 
^2         ' 


*  Professor  Chrystal  remarks:  "The  metlioil  was  probably  obtained 
by  analogy  from  the  arithmetical  process.  It  was  first  given  by  Recorde 
in  The  Whetstone  of  Witte  (black  letter,  1557)  the  earliest  English  work 
on  algebra."  It  would  be  serviceable  to  the  student  to  compare  the 
difference  between  the  numerical  and  the  algebraic  ijroblems. 


184  NUMBER   AND   ITS   ALGEBRA. 

To  find  c  we  might  have  taken  either  of  the  last  three 
coefficients. 

A  similar  method  would  yield  the  cube  root  of  a  function 
which  is  a  "  perfect  cube,"  etc. 


291.  Without  going  too  far  into  the  subject,  it  is  proper 
to  add  here  several  fundamental  theorems  concerning  com- 
plex numbers,  postponed  from  Chapter  XII. 

If  <f>  (x  4"  yi)  be  an  integral  function  of  a  complex  num- 
ber, we  saw  in  Chapter  XII.  that  it  is  reducible  to  a  com- 
plex number,  say  A  -j-  BL  Now,  if  all  the  coefficients  of 
^  (x  -{-  yi)  are  protomonic,  A  and  B  are  protomonic,  and  A 
can  contain  only  even,  and  B  only  odd,  powers  of  y  ;  there- 
fore, if  X  -\-  yi  be  changed  to  x  —  yi,  A  will  remain  unal- 
tered, and  B  changed  to  —  B.  That  is  to  say,  if  (|)  {x  -{- 
yi)  =  A  -\-  Bi,  ^  (x  —  yi)  =  A  —  Bi. 

The  theorem  is  readily  extended  to  include  all  stirpal 
functions,  integral  or  fractional,  of  a  complex  number,  and 
generalized  for  such  functions  of  more  than  one  complex 
number. 

292.  As  a  corollary,  if  all  the  coefficients  of  the  stirpal 
function  ^  (ii)  be  protomonic,  and  if  ^  (u)  =  0,  when  ?*  = 
a  +  bi,  then  ^  (u)  =  0,  when  u  =  a  —  bi;  for  if  ^  -|-  -^'<-  = 
0,  ^  =  0  and  ^  =  0  (§  193). 

State  the  corollary  for  ^  (u,  v,  iv  .   .  .). 

293.  Since  <|)  (x  -f  yi)  =  A  -{-  Bi  and  ^  (x  —  yi)  =  A 
—  Bi,  when  all  the  coefficients  in  the  functions  are  proto- 
monic ;  and  since 

norm  4,  (x  +  y  i)  =^  norm  {A  +  Bi)  =  A"-\-B-=  (A  +  Bi) 
(A  —  Bi)  ;    therefore 
norm  <^  (x  -\-  yi)  =  norm  ^(x  —  yi)  =  ^  (.«  -(-  yi)  (f)  (x  —  yi)  ; 
and  therefore  :  — 


COMPLEX   NUMBERS.  185 


mod  <f)(x  -\-  yi)  =  mod  <^ (re  —  y?")  =  -f-  V<^ (x  +  yi)  (j> (x  —yl) ; 

and  in  general 

mod  ^  [x  +  yi,  u  -\-  v'l,  .  .  .^  =  mod  <^{x  —  yl,  ?<  —  vi,  .  .  .  } 

=  +  V  {(/)  (a-  +  2/^',    n  +  '(;/,    .  ■.    .  )  (fi(x  —  yl,    w 

-t-.-.   .   .  )}. 

294.  If  the  function  be  the  product  of  several  complex 
numbers,  this  theorem  gives 

mod{(r+  si)  (t  -\-  ui){y  -\-  u-lj)  =  V{(^"+  si)  (t  +  nl)  (v  -\-  n'l) 
(r  —si){t  —in)(v  —  2rl)}  =  +  V{{r-~\-s-)(t'-\-u^)(v^-\-tv^)} 
=  V/'^  +  s'  ^/t'^  +  u-  Vv-  -\-  w'^  —  mod  l^r  -\-  si)  mod 
(t  -\-  III)  mod  (y  +  wi)  ; 

that  is  to  say,  the  modulus  of  the  product  of  any  number 
of  complex  numbers  equals  the  product  of  their  moduli. 

It  might  plausibly  be  taken  for  granted  (since  we  have 
seen  that  it  x  -\-  yi  =  0,  x  =  0,  and  ?/  =  0)  ;  but  it  is  better 
to  prove  distinctly  that  the  product  of  two  complex  num- 
bers cannot  be  zero,  unless  one  of  the  complex  numbers  is 
zero  : 

If  yz  =  0,  where  y  and  z  are  complex  numbers,  mod  (yz) 
=  0.  But  mod  (yz)  =  mod  y  mod  z  ;  therefore,  mod  y  mod 
z  =  0. 

But  mod  y  and  mod  z  are  protomonic.  Therefore,  either 
mod  y  =  0,  or  mod  z  =  0 ;  and  consequently,  by  Section 
198,   either  y  =  0,  or  z  =  0. 

295.  Again,  as  a  special  case  of  the  general  theorem  in 
Section  293,  if  the  particular  function  be  the  quotient  of 
two  complex  numbers,  we  have 

mod   S   ^  +  "\  \^+J\  ^+"\  .  ^  -  '"' 
I  V  -{-  tvl  )  \    I  V  -\-  ivl     V  —  u-i 

Y    I  v-'-{-  'W-'  \        Vw'  +  W       mod  (v  +  wl) 


186  NUMBER   AND   ITS   ALGEBRA. 

that  is  to  say,  the  modulus  of  the  quotient  of  two  complex 
numbers  is  the  quotient  of  their  moduli. 

296.  The  modulus  of  the  sum  of  complex  numbers  may 
equal  the  sum  of  their  moduli,  cannot  be  greater,  and 
is  in  general  less.  For  consider  two  complex  numbers, 
t  -f-  «^  and  v  -f-  iH. 

By  Section  293,  mod  (t  -\-  ul  -[-  v  -{-  vi)  = 

-\-  s/  {{t  -\-  ui  -}-  V  -\-  wi)  (t  —  ui  -\-v  —  u-i)}  = 
+  ^  {(*  +  ^)^  +  (^  +  '^y^}}  therefore  we  desire  to  prove, 
+  ^{(t-\-vy-\-{u-^wy}  not  >  +  V(^^+<)  +  V(z;^+w;2), 
or,  since  only  positive  roots  are  concerned,  that  (t  +  i.')^  + 
(u  +  tvf  not  >  t^-\-  u'  +  V-  +  IV-  +  2  V{b-'  +  u')  (v'^  +  w^). 
Subtracting  t'  +  n^  -{-  v"  -\-  n-^  from  both  members  of  this 
inequality  gives,  2  fv  -\-  2  mv  not  >  2  V(^^  +  u^)  (v'^  +  w"^), 
and  dividing  by  2,  tv  -\-  uw  not  >  -\J  {^^  +  m^)  (v'^  -f  it--). 

The  right-hand  member  is  essentially  positive,  and  there- 
fore not  less  than  the  left,  if  the  latter  is  negative  (as 
might  be  on  accomit  of  the  original  quality  of  t,  n,  v, 
or  ir)  ;  and  the  theorem  is  consequently  proved  for  that 
case. 

If  the  left-hand  member  is  not  negative,  by  squaring 
both  sides*  we  get 

t-v-  +  2  tuviv  -f  iihv'^  not  >  f-r"  +  nhr"  +  fic^  +  v^u^, 
or  2  tvwu  not  >  t'-^u-"  -\-  v-i(% 

or  0  not  >  t-ir-  +  r'~n-  —  2  tvwu, 

or  0  not  >  (fir  —  ?';/)-. 

But  this  is  true,  since  the  right-hand  member  is  essen- 
tially positive. 

297.  Argand's  diagram  beautifully  apjilies  to  geometri- 
cal relations  these  properties  of  complex  numbers,  thus 
analytically  displayed. 


THEOKY   OF   NUMBERS.  187 

•  298.  It  would  be  interesting  and  instructive  to  follow 
a  great  many  very  curious  and  useful  investigations  of 
various  properties  of  primary  or  discrete  (to  say  nothing 
of  complex,  or  continuous)  number^of  which  no  mention 
ever  has  been  made.  But  to  do  so  Avould  carry  us  into 
ideas  and  notations  equally  strange,  and  would  be  deemed 
a  transgression  of  appropriate  bounds  for  such  an  elemen- 
tary treatise  as  is  this  little  work.  For  instance,  Gauss 
makes  the  notion  of  congruence  fundamental  in  his  Disqui- 
sitiones  Avithmeticae,  Congruence  meaning  the  relation  of 
I  and  J,  if  7  =  ayx  +  r,  and  J  =^  h^-\-  r,  where  [x.  is  termed 
the  modulus  of  /  and  J,  and  /  and  J  are  called  congruent 
with  respect  to  modulus  /a.  Some  astonishing  facts  are 
directly  deducible  from  this  simple  mode  of  classification. 
It  is  not  from  the  difficulties  of  the  more  elementary  por- 
tion of  the  Theory  of  Numbers  *  that  the  field  lies  fallow 
for  our  undergraduate  courses  in  mathematics,  and  I  be- 
lieve the  interest  of  students  would  be  less  disposed  to 
flag  if  the  firmer  grasp  of  thought  were  commanded  Avliich. 
such  studies  would  infallibly  encourage. 


299.  If  the  equation  ^  (x,  y,  z)  =  ifr  (x,  y,  z)  is  satisfied 
for  all  values  of  the  variables,  it  is  called  an  identical 
equation,  or  an  identity,  or  a  formula.     (  Vide  §  40.) 

In  this  case  the  equation  is  formally  true,  under  the 
very  laws  of  numerical  operation,  regardless  of  particular 
values  of  the  variables. 

If,  on  the  other  hand,  an  equation  is  satisfied  only  for 
special  values  of  the  variables,  it  is  called  a  synthetic,  or 
conditional,  equation.     From  this  point  of  view,  the  con- 

*  For  bibliography  of  tlie  interesting  and  important  subject  which 
bears  tliis  name,  see  Numbers,  Theory  of,  Cayley,  Ency.  Brit.,  9tli  eil. 


188  NUMBER   AND   ITS   ALGEBRA. 

stants  are  commonly  spoken  of  as  known,  and  the  varitl- 
bles  as  unknown  "quantities,"  —  numbers,  in  the  algebra 
of  number. 

Synthetic  equations  are  classified  and  named  with  refer- 
ence to  their  unknown  numbers,  precisely  as  functions  are 
characterized  in  regard  to  their  variables.     {Vide  §  169.) 

Synthetic  equations  involving  only  stirpal  and  radi- 
cal functions  (exponential,  etc.,  equations  are  deferred  to 
future  studies)  can  always  be  made  to  depend  upon  an 
equation  of  the  form 

<\>  (a-,  y,  z,  .  .  .  )  =  Q, 
where  (/>  is  an  integral  function. 

This  form,  therefore,  is  of  prime  importance  in  the 
theory  of  equations. 

300.  Synthetic  equations  concerning  the  same  variables 
may  occur  in  sets,  or  systems.  In  this  case  they  are 
called  simultaneous,  and  the  problem  is  to  find  the  sets  of 
values  of  the  variables  which  render  every  equation  of  the 

.  system  an  identity. 

Such  a  set  of  values  is  said  to  satisfy  the  system,  and 
is  called  a  solution  of  the  system. 

Such  solutions  are  to  be  distinguished  in  many  ways 
from  the  solutions  of  one  integral  equation  in  one  variable, 
where  a  solution  is  called  a  root. 

301.  It  is  important  to  distinguish  between  two  differ- 
ent kinds  of  solution :  —  (1)  Numerical  solution,  exact  or 
approximate,  which  can  often  be  obtained  where  formal 
algebraical  solution  would  be  out  of  the  question ;  and  (2) 
What  may  be  called  formal  solution,  that  is,  a  solution  in 
which  the  variables  are  expressed  as  definite  analytical 
functions  of  the  constants.  Such  solutions  of  equations  of 
degree  higher  than  the  fourth  cannot,  in  general,  be  found. 


THEORY    OF    EQUATIONS.  189 

302.  The  final  test  of  any  solution  is  the  satisfaction 
of  the  equation,  upon  substitution  therein  of  the  values 
obtained  for  the  unknown  numbers.  No  matter  how  the 
solution  has  been  obtained,  if  it  does  not  stand  this  test, 
it  is  no  solution  ;  and  no  matter  how  obtained,  if  it  does 
stand  this  test,  it  is  a  solution.  It  is  often  a  good  way  to 
guess  a  solution,  and  make  the  test. 

303.  FUK^DAMEKTAL     PkOPOSITIOX     IX     THE    TlIEORY    OF 

Equations.  —  If  in  the  equation  <j),^  (x)  =  0,  (/>„  (x)  be  an 
integral  function  of  x  of  the  ?ith  degree  (the  coefficients,  in 
general  complex,  in  particular,  protomonic,  numbers)  where 
the  coefficient  of  the  a;"  term  is  not  zero,  then  ^„  (x)  is  the 
product  of  n  factors,  each  of  the  first  degree. 

With  one  provision  we  proved  this  proposition  in  Section 
268,  and  it  has  also  been  shown  that  these  factors  can 
always  be  in  the  form 

C  (x  —  r{)  (x  —  ?-2)  (a-  —  Ts)  .   .  .   (x  —  ?■„), 

where   C  is  the  coefficient   of  a:"  in  (^,^  (x),  and  Vi,  ?-2,  r^, 

...?•„  are  the  roots  of  the  equation.     Consequently,  the 

problem  of  solving  an  integral  equation  with  one  unknown 

number,   is    identical   with   the   problem   of   resolving  the 

general  function  of  one  variable,  of  like  degree,  into  factors 

of  form 

C  (x  —  ?-j)  (x  —  r^)  (x  —  vs)  .  .  .   {x  —  ?■„). 

304.  It  is  worth  while  to  call  attention  to  the  fact  that 

x^  -\-  X  -{- 1  =  (x  -{-  1  -\-  Va-)  (x  -\-l  —  Va-), 

often  given  by  beginners  when  required  to  factor  x'^  ~{-  x  -\-  1, 
although  a  true  identity,  is  no  factorization  in  the  sense 
intended,  because  the  factors  are  not  integral  functions. 

305.  ISTothing  need  be  said  of  the  solution  of  integral 


190  ISrUMBEK    AND   ITS    ALGEBRA. 

equations    of   the   first    degree :    properly  associating   the 

terms,  and  reducing  by  the  distril^utive  law  to  the  form 

iV" 
Cx  =  N.  gives  X  =  — . 

306.  Eecurring  to  Section  271  (4),  we  know  that 

(x  —  a)  (x  —  h){x  —  c)...   {x  —  n)  =  0 

is  an  equation  whose  roots  are  a,  h,  c,  .  .  .  n. 

Performing  the  multiplications,  we  have  the  form : 

a.»  +  r^.r«-i  +  r„.r"-^+    .    .    .    +r-„_j  a- +  c„  =  0, 
where,        c^  =  —  {a,  J^  h  -{-  c  -\-  .   .  .   -\-  n) 
Co  =  ab  -f-  "<'  -{-  he  -\-    .    .    .    -\-  vin 
Cz=  —  {'ihe  4-  uhd  -\-  acd  -[-.,.    -)-  liiui' 
r„  =  -j-  ahrd   .    .    .   n. 
(Plus  or  minus,  as  7i  is  even  or  odd). 

Hence,  if  an  integral  equation  of  the  ?ith  degree  is  in 
the  above  general  form  : 

The  coefficient  of  the  second  term  is  minus  the  sum  of 
the  roots. 

The  coefficient  of  the  third  term  is  the  sum  of  their 
products,  taken  two  at  a  time. 

The  coefficient  of  the  fourth  term  is  minus  the  sum  of 
their  products,  taken'  three  at  a  time,  etc. 

The  last  term  (the  constant)  is  plus  or  minus  the  product 
of  all  the  roots,  according  as  w  is  even  or  odd. 

307.  It  follows  :  In  every  equation  of  the  nth.  degree 
in  the  general  form, 

If  the  second  term  is  wanting,  the  sum  of  the  roots  is 

I  the  last  term  is  wanting,  at  least  one  root  is  zero, 
[f  all  the  roots  are  integral,  they  are  submultiples  of 
tiie   last  term,  which  must  be  integral.     But  the  inverse 


THEORY'    OF    EQUATIONS.  191 

does  not  follow ;  siuce  tlie  last  term  may  be  integral,  yet 
roots  be  fractional.  But  if  the  last  term  is  not  integral, 
some  of  the  roots  are  not  integral. 

If  all  but  one  of  the  roots  are  known,  the  remaining  one 
may  be  found  by  adding  the  sum  of  the  known  roots  to 
the  coefficient  of  the  second  term,  and  changing  the  quali- 
tative sign  of  the  result.  Or,  by  dividing  the  last  term  by 
l^lus  or  minus  the  product  of  the  known  roots,  according 
as  71  is  even  or  odd. 

If  m  roots  are  known,  the  equation  may  be  depressed 
to  another  of  the  {ii  —  m)th  degree,  by  dividing  by  the 
product  of  lit  factors  of  the  form, 

(x  —  7\)  (x  —  Vo)  .   .  .   (x  —  ''m)j  ^^^^  therefore  :  — 

If  all  but  two  roots  are  known,  the  coefficient  of  the 
depressed  equation  is  the  sum  of  the  known  roots  and  the 
coefficient  of  the  second  term  of  the  given  equation.  And 
the  last  term  of  the  depressed  equation  is  the  last  term  of 
the  given  equation,  divided  by  plus  or  minus  the  product 
of  the  known  roots,  according  as  n  is  even  or  odd. 

308.  From  the  process  of  multiplication  required  in  Sec- 
tion 306,  it  is  evident  that  if  all  the  r's  are  positive,  the 
quality  of  the  terms  is  alternately  +  ^^^^  —  •  Hence,  if 
the  roots  of  an  equation  are  all  positive,  the  signs  of  its 
terms  (supplying  missing  terms  by  zeros)  are  alternately 
-J-  and  — ,  and  inversely. 

Again,  if  all  the  r  's  be  negative,  there  is  no  change  in 
the  signs  of  the  terms. 

It  would  not  be  difficult  to  deduce  here  Descarte's  Rule 
of  Signs :  An  integral  equation  cannot  have  more  positive 
roots  than  it  has  changes  of  signs,  nor  more  negative  roots 
than  it  has  continuations  of  the  same  sign. 


102  NUMBER    AND    ITS    ALGEBRA. 

309.  Prove  :  Any  integral  equation  may  be  transformed 
into  another  whose  roots  are  the  negatives  of  the  original 
roots,  by  changing  the  signs  of  alternate  terms,  beginning 
with  the  second. 

310.  To  transform  an  integral  equation  into  another, 
whose  roots  are  the  roots  of  the  original  equation  multi- 
plied by  a  given  number,  k :  — 

In  tlie  general  form  substitute  y  /  /-  for  x,  obtaining,  — 


Multiplying  by  k"-  gives 
//"  +  ^iAr-'  +  ^./.y--+  .  .  .r„_i7.-«-V+^„A-  =  0  (2) 
The  roots  of  (2)  are  the  values  of  ?/  that  satisfy  it;  but 
1/  =  kx ;  therefore,  noting  the  coefficients  in  (2),  to  effect 
the  desired  transformation,  multiply  the  second  term  by  k, 
the  third  by  k'-^,  and  so  on. 

311.  Equations  may  be  transformed  in  many  other  use- 
ful ways  ;  for  example,  so  that  the  roots  shall  be  the  ori- 
ginal roots  ^  some  constant.  This  mode  of  transformation 
is  most  serviceable  in  the  special  case  of  making  the  exact 
increment  which  will  cause  the  second  term  to  vanish,  —  a 
device  for  preparing  cubic  and  biquadratic  equations  for 
solution.     For  a  simple  illustration  see  Section  316. 

312.  Seeing  that  we  have  the  unique  resolution  :  ■ — - 

^u  (^-)  =  0  =  c(x  —  ?-j)  (x  -  ;•.,)   .   .  .   (x  -  ?•„), 
it  follows  from  Section  292  that  if  ^„{x)  has  all  its  coeffi- 
cients protomonic,  and  vanishes  when  x  =  a-{-  hi,  it  must 
vanish  when  x  =  a  —  hi. 

This  is  to  say,  that  in  any  integral  equation  whose 
coefficients  are  protomonic,  roots  which  are  complex  num- 
bers must  occur  in  conjugate  pairs. 


QUADRATIC    EQUATIONS,  193 

In  like  manner  {vide  §  170)  surd  roots  can  enter  equa- 
tions with  commensurable  coefficients  only  in  conjugate 
pairs. 

Thus,  all  such  equations,  if  of  an  odd  degree,  must  have, 
in  the  former  case  at  least  one  protomonic,  and  in  the 
latter  at  least  one  commensurable,  root. 

313.  The  general  equation  of  the  second  degree  in  one 
variable  is  ax^  +  hx  -(-  ^  =  0.  The  general  theory  of  solu- 
tion is  already  in  our  hands,  and  in  this  case  the  formal 
solution  (vide  §  301)  is  always  obtainable.  Various  methods 
may  be  followed. 

The  general  equation, 

ax^  -f-  ^■^-"  +  c  =  Oj 
may  be  reduced  without  altering  the  roots  (§  303)  to 

a  a 

or  X-  -\ —  x  = . 

a  a 

From  consideration  of  the  formula  (x  -\-  yy  z=  x-  -\-2  xy 
-f-  ij'\  it  is  plain  that  the  left-hand  member  may  be  made  a 

/    h    \2 

''  complete  square  "  in  x  by  adding  f  7y~~  )   to  each  member, 
which  gives  — 

X'^  -\ X  -\ = 1 =  . 

a         4  a^  a       4  a^  4  a^ 

Taking  the  square  root  of  each  member,  * 

*  The  double  sign  before  the  left-hand  member  would  be  superfluous, 
since  nothing  more  would  be  said  than  is  expressed  as  the  statement 
stands;  e.g. :  — 

±  (a  -\-  h)  —  ^  {c  -\-  d)  says  no  more  than 

a  -f  ?>  =  ±  {c-^  d),  as  one  may  readily  satisfy  himself. 
See  also  Section  325.  _„.»—__ 


194  NUMBER,    AND    ITS    ALGEBRA. 

X  +  ^-^  =  ±  .— V^/'  -  4  ae, 
la  la 

or  3.  ^  Z-IL  i  "^^'''  —  4  «c 


2  a 


We  have  here  a  formal  solution  of  the  general  quadratic 
equation. 

Also  the  quadratic  function,  ax^  -\- hx  -[-  c,  has  been  fac- 
tored. For,  by  the  principles  clearly  exhibited  in  Section 
303,  —  

ax--\-bx-\-c=^n{  x ■ — ■ "  -' 


2  a  J 

314.  In  solving  a  particular  quadratic  in  one  variable, 
we  may  give  this  process  of  *'  completing  the  square  "  its 
particular  application ;  or  we  may  employ  the  formal  solu- 
tion as  a  rule;  that  is,  after  reducing  the  given  equation 
to  the  form  ax"  -)-  /y^  -|-  c  =  0,  simply  write  down  the  partic- 
ular values  in 


X 


—  —  ^  jz  V^'  -  4  ac 


2  a 


Of  course,  if  the  given  equation  in  form  ax-  -{-  hx  -\-  c  = 
0,  affords  a  function  readily  factorable  by  inspection,  it 
would  be  absurd  to  feign  an  investigation  for  what  is 
already  known.  For  instance,  one  with  any  skill  in  the 
algebra  cannot  fail  to  see  that  in  a-^  +  5  a;  -f  G  =  0,  we 
have  (x  +  3)  (a;  -f  2)  =  0  ;  which  is  to  say,  that  x  =  —  3, 
and  X  =  —  2. 

The  device  of  reducing  the  given  equation  to  the 
form,  4  a'^x-  -\-  4  ahx  +  4  ac  =  0,  before  "completing  the 
square"  (known  as  the  Hindoo  Method),  is  hardly  Avorth 
mentioning,  since  it  merely  avoids  fractions  which  offer  no 
obstacle  to  calculation.  It  is  doubtless  a  relic  of  the  times 
when  fractional  number  was  regarded  with  suspicion. 


HOOTS    OF   QUADRATIC    EQUATIONS.  195 

315.     If  i\\e  formal  solution  of  ax-  -\-  hx  -\-  c  ^  0, 


—  h  A-  -Vb-  —  4  ac 
X  = ^ , 

2a 

be  coDsidered,  it  will  be  seen  that,  when  the  coefficients  are 
protomonic,  the  roots  are  :  • — • 

(1)  Protomonic  and  unequal,  if  Ir  —  4  ac  is  positive. 

(2)  Protomonic  and  equal,  if  //-  —  4  ac  =  0. 

(3)  Commensurable,  if  V^"  —  4  ac  is  commensurable. 

(4)  Conjugate  surds,  if  V^-  —  4  ac  is  incommensurable. 

(5)  Conjugate  complex  numbers,  if  ^-  —  4  ac  is  negative. 

(6)  Equal,  if  //-  =  4  ac. 

(7)  Equal  moduli,  but  one  positive,  other  negative,  if 


as   —   -  is  +  or  —  ] . 


protomonic  or  neomonic, 


(8)  One  zero,  other  =  —  h/  a,  if  c  =  0. 

(9)  Both  zero,  it  b  =  0  and  c  =  0. 

It  may  be  profitable  to  find,  from  a  different  standpoint, 
more  or  less  the  same  criteria :  — • 

From  Section  306,  the  equation,  ax'  -j-  bx  -\-  c  ^  0,  gives 
the  following  relations  of  roots  and  coefficients, — 

r^  -(-?•„  = ,    and    /^  ?'o  ==  -  . 

a  'a 

Consequently  1\  and  r^  are 

.  ."     .     b  ■  .  c 

positive  if  -  is  negative  and  -  positive ; 

a  a 

negative  if  -  is  positive  and  -  positive ; 
a  a 


196 


NUMBEE    AND    ITS    ALGEBKA. 


of  opposite  quality  if  -  is  negative. 


a 


Tliese  statements  presuppose  (1)  above,  ^-  —  4  ac  >  0. 


ri  =  —  To   if 


0. 


a 


ri  =  0    or    ?-o  =  0    if   -  =  0. 

a 

ri  =  0    and    r.  =  0    if   -  =  0    and    -  =-  0. 

'  a  a 

If  ax^  -|-  ix  +  c  =  0  be  still  regarded  as  a  quadratic 
when  a  =  0,  then  one  root  is  co  .  If  ^  also  is  zero,  both 
roots  become  infinite. 

These  criteria  may  be  tabulated  :  — 


KouTs. 

Ckitekiox. 

]{OOT.S. 

Ckiteiuon. 

Protomonic    .    .    . 
Commensurable     . 

Surd 

Complex     .... 

Equal 

Equal  moduli,  but 
one +,  other—    . 

62  —  4 ac>  0. 

Positive    .    . 
Negative  .    . 

One+,  One  — 

One,  0    .     .     . 
Both,  0      .     . 
One,  CO  .     .     . 
Both,  00     .     . 

^+,and^-. 
a               a 

^+,and^+. 
a              a 

c  __ 

a 

c  =  0. 

6  =  0  and  c  =  0. 

a  =  0. 

rt  =  0  and  6  =  0. 

i/b-  —  iac,  commen- 
surable. 

V'62  — -lac,  surd. 
0-  —  4ac  <0. 
62  —  4  ac  =  0. 

6  =  0. 

316.  Another  method  of  solving  a  quadratic  equation 
is  important  from  its  bearing  on  the  solution  of  cubic 
equations. 

The  general  equation,  ax^  -\-  bx  -\-  c  =  0  .  .  .  (1),  may  be 
reduced  by  a  change  in  the  variable  to  the  form  ai/^  -|-  (Z  = 
0  .  .  .  (2),  from  the  immediate  solution  of  Afhich  the  origi- 
nal variable  is  recovered.  To  discover  Avhat  change  must 
be  made  in  the  original  to  serve  this  purpose  (^vide  §  311), 
let  X  —  1/  -\-  e. 


INDETERMINATE    SYSTEMS.  197 

Ji  X  =  7/  -{-  e,  (1)  is  equivalent  to 

«  (1/  +  ey-i-b(y  +  e)  +  c  =  0; 
or  of?/2  ^  (2  ae  -\-  b)  y  -\-  ae"-  -\-  he  -\-  c  =  0.  (3) 

To  make  the  second  term  vanish,  2  ae  -\-  b  must  be  zero, 

b 

or  e  = 


2  a 

With  this 

vah;e  of  e,  (3)  becomes 

<,        b-  —  4:  ae        ^ 
4  a 

whence 

±■^/b^  -4=  ac 
^                 2a 

But 

x  =  y  -\-  e  =  y  -  --; 

2  a 

therefore 

—  Z*  ±  V/'^  —  4  ac         ,    r 
X  = ,  as  before. 

2a 

317.  If  an  equation  contaifis  two  {a  fortiori,  more  than 
two)  unknown  numbers,  it  is  obviously  indeterminate.  An 
extraneous  condition  (e.g.,  that  the  variables  shall  be  in- 
tegers) sometimes  affords  a  basis  for  a  determinate  solution, 

A  system  of  simultaneous  equations  is  in  general  deter- 
minate when  the  number  of  equations  equals  the  number 
of  the  variables. 

If  the  number  of  equations  is  less  than  the  number  of 
variables,  the  solution  is  in  general  indeterminate. 

If  the  number  of  equations  is  greater  than  the  number 
of  variables,  there  is  in  general  no  solution,  the  system 
being  inconsistent,  contradictory. 

These  are  ultimate  logical  principles ;  special  limitations 
of  the  statements  are  needed  rather  than  proof. 

It  must  suffice  here  to  point  out  that  a  system  may  be 
apparently  determinate,  yet  indeterminate  by  reason  of  one 


198  NUMBER   AND  ITS   ALGEBRA. 

being  analytically  derivable  from  the  others.  Also  it  may 
happen  that  a  system  of  analytically  independent  equations 
may  have  more  equations  than  variables,  yet  not  be  con- 
tradictory. 

Let  the  student  frame  examples  of  such  conditions. 

318.  A  determinate  system  of  integral  equations  involv- 
ing the  variables,  x,  y,  z,  .  .  . ,  cannot  have  more  than,  and 
in  general  has  exactly  abc  .  .  .  solutions,  where  a,  b,c,  .  .  . , 
are  the  degrees  of  the  system  in  the  respective  variables. 

Proof  of  this  proposition  must  await  future  studies  ;  but 
it  is  useful  to  know  the  theorem,  and  the  question  presents 
itself  at  once,  and  should  not  be  ignored  by  the  teacher. 

319.  Two  systems  of  equations,  each  of  which  may  con- 
sist of  only  one,  are  termed  equivalent  when  every  solution 
of  each  is  a  solution  of  the  other. 

From  any  system  we  may,  in  an  infinite  variety  of  ways, 
deduce  another  system ;  but  the  derived  system  is  not  gen- 
erally equivalent  to  the  original. 

This  matter  is  of  fundamental  importance,  even  at  the 
most  elementary  stages.  It  is  commonly  (with  several 
notable  exceptions)  left  in  the  dark  by  our  text-books, 
though  "  there  are  few  parts  of  algebra  more  important 
than  the  logic  of  the  derivation  of  equations,  and  few,  un- 
happily, that  are  treated  in  more  slovenly  fashion  in  elemen- 
tary teaching.  Xo  mere  blind  adherence  to  set  rules  will 
avail  in  this  matter  ;  while  a  little  attention  to  a  few  simple 
principles  will  readily  remove  all  difficulty."  * 

320.  If  A  and  B  are  two  functions,  which  do  not  become 
infinite  for  any  finite  values  of  the  variables  (such  cases 
must  be  considered  separately),  the  only  values  of  the  vari- 

*  Text  Book  of  Algebra,  Chrystal,  vol.  i.,  p.  285. 


EQUIVALENCE    OP    EQUATIONS.  199 

ables  which  make  ^  x  ^  =  0  are  such  as  make  yl  =  0,  or 
^  =  0,  according  to  laws  already  fully  demonstrated. 

321.  Axiomatically,  ii  A  =  I>,  (1) 
then                              A-^C=  B  ^  C.  (2) 

Also,  (1)  and  (2)  are  equivalent,  for  neither  can  be  true 
xmless  the  other  is  true. 

Note  the  corollaries  whereby  we  "  transpose  a  term  with 
changed  signs,"  or  "  change  all  signs,"  or  reduce  any  equa- 
tion to  the  form  ()  =  0,  without  destroying  equivalence. 

322.  On  the  other  hand,  although,  if 

A  =  B,  (1) 

then  AC=BC,  '  (2) 

the  derivation  being  perfectly  legitimate,  and  the  resulting 
equation  true,  yet  (2)  is  not  equivalent  to  (1),  unless  C  is 
a  constant  not  zero ;  for,  by  Section  321  (2)  is  equivalent 

^°  AC-BC=0 

that  is  to  C(A-  B)  =0  (3) 

Now,  if  C  is  a  constant  not  zero,  (3)  is  equivalent  to  (1)  by 
Section  320 ;  but  not  otherwise,  for  if  C  is  a  function  of 
the  variables,  (3)  is  satisfied  by  all  values  of  the  variables 
that  satisfy  the  equation,  C  =  0  .  .  .  (4),  which  in  general 
will  not  satisfy  (1).  Therefore  (2)  is  not  equivalent  to 
(1),  but  to  (1)  and  (4). 

In  this  way  it  is  plain  that  multiplying  both  members  of 
an  integral  equation  by  an  integral  function  introduces 
roots,  and  dividing  the  members  of  such  an  equation  by 
an  integral  function  loses  roots. 

Also,  from  any  integral  equation  another  equivalent 
equation  can  always  be  derived  in  which  the  coefficient  of 
any  term  shall  be  as  desired,  say  -f-  1  for  the  highest  term ; 
for  this  is  obtainable  by  multiplying  by  a  constant. 


200  NUMBEK   AND   ITS   ALGEBRA. 

323.  Fractional  equations  must  never  be  confounded, 
in  the  matter  of  degree  and  number  of  roots,  with,  integral 
equations.  The  very  term  degree  does  not  apply  to  frac- 
tional equations.     Fractional  functions  of  x  may  sometimes 

be  integral  functions  of  some  function  of  .r  f  e.g.,  -  j;  but 

in  general  no  sucli  relations  as  obtain  between  degree  and 
number  of  roots  in  integral  equations  subsist  for  fractional 
equations.  The  latter  must  be  solved  under  the  logic  of 
the  equivalence  of  derived  integral  equations. 

From  any  fractional  equation  an  integral  equation  may 
be  deduced,  which  may  or  may  not  be  equivalent.  If 
E  ^=  F,  where  E  and  F  are  fractional  functions,  and 
Jf  =  1.  c.  m.  of  the  denominators  in  E  and  F,  then  EM  = 
FM  is  integral. 

Here  extraneous  solutions  of  M  =  0  may  be  introduced, 
but  not  necessarily  or  generally.  E  and  F  contain  frac- 
tions whose  denominators  are  factors  in  31,  and  in  general 
roots  of  M  =0  would  make  E  ov  F  infinite,  and  conse- 
quently M  (E  —  F)  not  necessarily  zero. 

See  examples  below  for  clear  understanding  of  this 
point. 

324.  If  both  members  of  an  equation  be  raised  to  the 
same  power,  in  general  the  resulting  equation  is  not  equiva- 
lent. Thus  A  =  B;  then  A^  =  B%  or  A^  -  B^  =  0.  But 
the  last  is  equivalent  to  {A  -\-  B)  {A  —  B)  =  0;  hence  the 
solutions  of  A  -\-  B  =  0  would  in  general  be  introduced. 

It  may  be  noted  that  in  squaring  A  =  B  the  result  is 
the  same  as  if  the  members  of  the  equivalent  equation, 
A-  B  =  0,  were  multiplied  by  A  +  B.     {Vide  §  322.) 

325.  Neither  the  equation  between  the  positive,  nor  that 
between  the  negative,  square  roots  of  the  members  of  the 


EQUIVALENCE   OF   EQUATIONS.  201 

equation  A  =  B,  is  an  equivalent  equation ;  but  the  two 
equations  (generally  written  together  with  double  signs) 
between  the  positive  root  of  one,  and  both  roots  of  the 
other,  constitute  an  equivalent  system.      (Vide  §  313.) 

+  VX=  +  V^  (1) 

nd  +  VA  =  -VB  (2) 

is  a  system  equivalent  to  ^  =  Z?. 

For  A  =  B  is  equivalent  to  A  —  B  =  0,  which  is  equiva- 
lent to  {-y/A  -\-  -y/B)  (VA  —  -y/B)  =  0,  which  is  equivalent 
to  the  system  (1)  and  (2). 

326.  li  A  =  B  he  Q,  radical  equation,  repeated  involu- 
tions Avill  deduce  an  integral  equation  which  may  or  may 
not  be  equivalent.  Extraneous  solutions  may  be  intro- 
duced ;  and,  if  like  roots  in  the  original  equation  alone  be 
regarded,  often  no  solution  of  the  derived  equation  will 
satisfy  the  original. 

327.  Two  equations  which  are  not  equivalent  are  called 
indejjendent.  Two  or  more  independent  equations  involv- 
ing a  corresponding  number  of  variables  may  be  capable  of 
coincident  solution  ;  if  so,  they  are  termed  simultaneous, 
that  is,  consistent,  or  involving  variables  which,  though  un- 
known, are  the  same.  Contradictory  statements,  no  matter 
how  artfully  veiled  the  contradiction,  can  lead  only  to  non- 
sense in  algebra,  as  elsewhere. 

Compare  again  Sections  300,  317,  318. 

The  devices  of  elimination,  whereby  an  equation  in  one 
variable  is  deduced  from  a  system  of  simultaneous  equa- 
tions in  several  variables,  are  familiar ;  but  the  logic  of 
such  derivations,  and  the  paramount  question  of  the  equiv- 
alence of  the  derived  and  original  systems  may  have  been 
overlooked. 


202  NUMBER   AND   ITS    ALGEBRA. 

The  present  discussion  must  be  concluded  with  two 
propositions  specially  concerning  the  equivalence  of  simul- 
taneous systems.  The  subject  will  have  been  by  no  means 
exhausted ;  but  my  purpose  of  stimulating  alert  and  intelli- 
gent observation  in  the  important  matter  of  solving  alge- 
braic equations  will  probably  be  fulfilled.  The  student's 
skill  and  knowledge  will  steadily  increase,  if  strict  atten- 
tion be  always  paid  to  the  question  of  equivalence. 

328.    The  system, 

P  ^^  n  S^^ !  I  i«  equivalent  to  ^  =  0  (1)  | 

for  any  solution  of  I  makes  A  zero,  and  B  zero,  and  there- 
fore satisfies  II;  and  any  solution  of  II  makes  A  zero,  and 
therefore  reduces  II  (2)  to  q  B  =  0,  ov  B  =  0. 

Conseqviently  any  solution  of  either  satisfies  both. 

It  may  be  suggestive  to  state  this  proposition  again  in 
the  form 

A  =  B)  A  —  P 

^  ?  is  equivalent  to 
C  =  D\  AA-C  =  B-\-D 

On  the  other  hand,  — 

^~  ^}I  is  not  equivalent  to     ^  =  ^     |  ,j 
C  =  B\  ^  AC  =  Bd] 

For,  though  all  the  solutions  of  I  are  solutions  of  II,  II 
has  in  addition  all  the  solutions  of  C  =  0,  and  D  =  0. 
Let  the  student  satisfy  himself  of  the  truth  of  this  propo- 
sition. It  explains  many  ''answers"  which  may  have 
been  incomprehensible  to  him. 

The  following  examples  may  serve  to  impress  what  has 
been  said  concerning  the  equivalence  of  derived  equations 
with  their  originals,  although  at  every  point  the  student 
should  have  found  specific  illustrations. 


EQUIVALENCE   OF   EQUATIONS.  203 

329.    (1)    Solve 

^  ^^    =1.  (1) 


X  —  3       X  -\-  (J 

Multiplying  each  member  by  (.r  —  3)  (.r  -f  G)  gives 
a-2  -3x  -  18  =  0, 
or,  (x  _  6)  (x  +  3)  =  0,  (2) 

whence  x  =  6  and  x  ^=  —  3. 

Both  of  these  are  solutions  of  (1).  i^o  roots  of  (x  —  3) 
(x  -j-  6)  =  0  were  introduced,  because  x  ^  3  or  x  =  —  6 
would  make  the  left-hand  member  of  (1)  infinite,  and 
therefore  M  {E  -  F)  not  zero.     {Cf.  §  323.) 

(2)  Solve         1 —  =  — 6.  (1) 

Transposing,  and  adding  the  fractions,  gives 

1  -  -^ J  =  -  6, 

X  —  1 

or  1  —  (x  +  1)  =  —  G, 

or  a-  =  G      .  .  .  equivalent  to  (1). 

But  a  beginner  might  multiply  by  a;  —  1,  deriving 

ic  -  1  -  a--^  =  -  1  -  G  a;  +  6  (2) 

whence  x  =  \  and  a*  =  G, 

where  1  is  no  solution  of  the  original,  and  therefore  (2)  is 
not  equivalent  to  (1). 

Multiplying  by  any  integral  function,  not  nesessary  to 
clear  of  fractions,  will  derive  an  equation  not  equivalent. 
Accordingly,  every  device  for  identical  simplification  should 
be  employed  before  multiplying  by  the  lowest  common 
multiple. 

(3)  Solve    i_^^''  +  ^-6^^^-3a-  +  2^  ^ 


204  NUMBER   AND   ITS    ALGEBRA. 

Multiplying  each  member  by  (x  —  2)  (x  +  2),  and  redu- 
cing identicall}-,  gives 

3  «2  _  4  a;  _  4  =  0,  (2) 

whence  x  =  ^  Jz  VI6  +  48  ^^_Al  =  2o.-  2/3. 

6  6  ' 

Equation  (2)  is  not  equivalent  to  (1),  the  root,  2,  of 
(x  —  2)  (x  4-  2)  =  0  having  been  introduced,  because  the 
fraction  in  the  left-hand  member  of  (1)  is  not  in  its  lowest 
terms.  If  (1)  be  reduced  before  clearing  of  fractions  we 
obtain 

14-0^  +  3^^  ~       o"^*^? 
X  -^  2 

whence,  multiplying  hj  x  -j-  2, 

x^  -\-  6  X  -\-  8  =  x^  -  3  X  -\-  2,  (3) 

or  X  =  -  2/3, 

where  (3)  is  equivalent  to  (1). 


(4)    Solve  V4  -  .T  =  ic  —  4.  (1) 

Squaring  4  —  x  =  x-  —  8  x  -\-  16, 

or  a"2  —  7  a-  4-  12  =  0, 

or  (x  -  3)  (x  -  4)  =  0,  (2) 

whence  x  =  S  and  x  =  4. 

Of  these  solutions  of  (2),  4  is  a  solution  of  (1)  if  the 
l^ositive  square  root  be  taken,  and  3  is  not  a  solution  ; 
whereas,  if  the  negative  root  be  taken,  3  is  a  solution  and 
4  is  not.     Thus  (2)  is  equivalent  to 


-\-  V4  —  X  =  X  —  4:  and  —  V4  —  x  =  x  —  4. 

(5)    Solve     V3rK-|-  1  =  V9a;  +  4  -  V2 x-l  (1) 

Squaring  twice,  and  reducing  identically,  gives 

a;2  _  I  a;  _  5  ==  0  (2) 


whence  x  =  5  and  x  =  —  ^ 


EQUIVALENCE   OF   EQUATIONS.  205 

Using  only  positive  roots  of  the  radicals,  5  is  a  solution  of 
(1) ;  but  —  ^  substituted  in  (1)  gives 

or  ^  V2  i  =  h  V2  i  —  V2  i,  (3) 

an   absurdity  if    the   statement  be   restricted   to   positive 
roots ;  but  if  the  negative  root  of  the  left-hand  member  be 
taken  with  the  positive  roots  of  the  terms  in  the  right- 
hand  member,    (3)  is  an  identity. 
Therefore  (2)  is  equivalent  to 

-f  VSu;  +  1  =  +  VOu;  -f  4  -  (+  V2  X  —  1), 


and      —  V3  a;  +  1  =  +  V9  ic  +  4  —  (+  V2  a;  —  1). 


(6)    Solve      2  -  V2  X  +  8  -f  2  v./;  +  5  =  0  (1) 

Squaring  twice^  we  deduce 

a-2  =  16  (2) 

whence  a;    =  -JL  4. 

In  this  case,  using  the  positive  roots  of  the  radicals  in 
(1),  neither  -|-  4  nor  —  4  is  a  solution. 

So  far  as  I  am  acquainted  with  them,  treatises  upon 
algebra,  if  they  notice  such  cases,  merely  declare  that  the 
original  equation  is  impossible  and  has  no  solution.  Pro- 
fessor Chrystal  states  the  theorem  :  — 

"  From  every  algebraical  equation  we  can  derive  a 
rational  integral  equation,  ivhlch  to'ill  he  satisfied  Inj  antj 
solution  of  the  given  equation  ;  but  it  does  not  follow  that 
every  solution,  or  even  that  any  solution,  of  the  derived 
equation   will  satisfy  the  original  one." 

The  italics  are  mine,  and  would  mark  logical  contradic- 


206  NUMBER   AND    ITS    ALGEBRA. 

tions  if  there  is  "  any  solution  "  of  the  original.     Professor 
Clirystal's  example  is  :  — 


V*  +  1  4-  V^'  —  1  =  1. 

The  derived  equation  yields  the  single  solution,  x  ~  & . 
The  only  remark  is,  "  it  happens  here  that  a;  =  |  is  not  a 
solution." 

Note,  I  is  a  solution  of  +  V*'  +  1  +  (—  V.e  —  1)  =  1. 

Professor  Taylor,  in  his  Academic  Algebra,  Boston,  1893, 
which  deserves  rare  praise  for  emphasizing  from  the  begin- 
ning the  question  of  equivalence  of  equations,  uses  example 
(6)  above,  concluding  "2  —  V2  x  +  8  +  2  Vx  +  5  =  0  is 
an  impossible  equation,  for  it  has  no  solution." 

Now,  I  must  not  be  understood  as  disputing  these  state- 
ments ;  they  are  true,  taking  the  numerical  statements  to 
be  restricted  to  positive  roots.  But  it  seems  to  nie  that 
the  student  stands  in  need  of  further  explanation :  he 
should  be  directed  to  observe,  that  though  one  may  write 
down  what  he  pleases,  as  an  isolated  statement,  no  restric- 
tions can  be  put  upon  the  operational  effect  of  such  nu- 
merical relations.  The  square  root  of  4,  as  an  inexorable 
fact,  is  -|-  2  or  —  2.  In  general  operation,  radical  surds 
necessarily  include  all  their  roots.  If  one  says,  V-*^,  he  has 
expressed  six  distinct  subjects  of  affirmation,  nor  can  the 
logical  consequences  of  these  alternatives  be  avoided  in 
numerical  analysis. 

The  conclusion  of  the  particular  problem  under  consid- 
eration is,  that  no  finite  number  satisfies  the  equation, 
taking  positive  square  roots ;  but  by  reason  of  the  perfect 
generality  and  freedom  of  numerical  operations,  if  there  is 
a  number  such  that  either  of  the  square  roots  concerned 
fulfils  the  conditions,  it  must  be  yielded  as  a  solution  of 


EQUIVALENCE    OF    EQUATIONS.  207 

the  equation.  We  had  occasion  to  notice  in  Section  323, 
and  in  example  (1)  above,  that  indeterminate  infinite  solu- 
tions do  not  obliterate  or  interfere  with  finite  solutions,  if 
there  be  any  such. 

And  in  general,  the  complete  analysis  of  any  radical 
equation  would  seem  to  require  the  investigation  of  all  the 
alternative  equations  arising  from  the  indifferent  roots  of 
radical  surds.  Some  of  these  niay  be  impossible,  in  the 
sense  of  having  no  finite  solution ;  but  if  a  finite  number 
will  satisfy  any  one  in  the  system,  it  will  certainly  discover 
itself  in  the  attempted  solution  of  an}'  other,  —  and  simply 
because  the  choice  of  particular  roots  is  arbitrary,  and  an 
equation  cannot  be  made  to  yield  nonsense,  or  contradic- 
tion, so  long  as  there  is  possible  consistency  of  its  terms. 

(7)    Solve  the  simultaneous  system 

x-2  +  2y2  =  9  (2)  f 

Solving  (1)  for  x  x  =  5  —  2>j  (3) 

Substituting  in  (2)  from  (3)  (5  -  2//)^  -(-  2i/  =  9. 

or  3y-  -  lOy  +  8  =  0 

or  (3y_4)(y-2)  =  0  •  (4) 

System  A  is  equivalent  to  system  B  (calling  (3)  and  (4) 
system  B).  But  system  B  is  equivalent  to  the  double 
system 

^  =  ^--^U,    and     ^  =  ^-^^1^ 
3y_4  =  0)  7/_2  =  0[ 

r'  A 

The  solution  of  c  is  a;  =  —,?/  =  —. 

3    -^       3 

The  solution  of  fZ  is  x  =  1,  ?/  =  2. 

Hence  these  are  the  two  solutions  of  A.     (Vide  §  318.) 


208  NUMBER    AND    ITS    ALGEBKA. 

(8)    Solve  the  simultaneous  system 

x^-2xy=      0  (1)  I 

4x2  ^  9^2  _  225  (2)  I  ^ 

Factor  (1)  x{x-  2y)  =  0. 

Hence  A  is  equivalent  to  the  double  system 

4^2  ^  92  =  225  ")  ,  ,     4*2  -L  9y2  _  225  ) 

'      -^  V  ^,    and  '      "^  -  c. 

a;  =  0       )  ic-2^  =  0      i 

The  solutions  of  •'^  are  obviously  a-  =  0,  ?/  =  5 ;  and  a*  = 
0,  ^  =  —  5.     Substituting  ;r  =  2//  in  the  first  equation  of 

'  ^^^^^  r  =  0,  or  y  =  ±  3. 

Substituting  in  a:  —  2//  =  0  we  have  for  the  solutions  of 
c,  X  ^  Q>,  1/  =  3 ;  and  x  =  —  6,  y  =  ~  3. 

Hence  the  four  (vide  §  318)  solutions  of  A  are  x  =  0, 
7/  =  5 ;  X  =  0,  7/  =  ~  5;  X  ^6,  7/  =  3;  X  =  —  6,7/  =  —3. 
In  the  solution  of  simultaneous  systems,  attention  must 
always  be  given  to  the  correct  association  of  values  of 
the  variables. 

330.  When  a  simultaneous  system  has  its  equations  of 
the  second  degree,  its  solution  demands  in  general  the 
solution  of  a  biquadratic  equation  in  one  variable.  Inas- 
much as  the  studies  to  which  these  lectures  are  intro- 
ductory may  be  regarded  as  beginning  about  at  this  point, 
I  bring  these  discussions  to  a  close,  without  treating  of 
the  solution  of  simultaneous  quadratic  systems,  or  of  cubic 
equations,  or  of  biquadratic  equations,  to  say  nothing  of 
equations  of  liigher  degree,  except  in  so  far  as  the  general 
fundamental  theory  may  suffice  in  particular  instances. 

Such  matters  are  to  be  studied  in  detail ;  but  it  may 
be  remarked  in  closing  tliat,  if  a  simultaneous  quadratic 
system  has  only  one  of  its  equations  of  the  second  degree, 
or  if  the  equations  are  homogeneous  or  symmetrical  (vide 


HIGHER    EQUATIONS.  200 

§  263),  means  are  offered  for  the  deduction  of  equivalent 
equations  in  one  variable  of  the  second  degree,  and  the 
system  may  in  these  cases  be  solved  by  the  methods  for 
quadratics.  Indeed,  it  is  often  the  case  that,  on  account  of 
symmetry,  this  is  true  for  a  system  of  simultaneous  equa- 
tions of  degree  higher  than  the  second.  Again,  any  equa- 
tion of  form,  aa;-"  +  hx^  -|-  c  =  0,  may  be  solved  as  a 
quadratic  in  cc",  and  the  two  unaffected  equations,  a;"  = 
-1-  k,  which  result  may  then  be  solved  by  factoring  the 
functions  a'"  -|-  k  and  a,-"  —  1:,  if  n  be  integral,  or  by  invo- 
lution of  the  members  of  x"-  =  -J-  k  if  n  be  fractional  with 
numerator  1,  or  by  both  devices  if  n  be  fractional  with 
numerator  >  1.  For  it  must  never  be  overlooked  that  the 
solution  of  an  integral  equation  in  one  variable,  in  form 
yi  =  0,  is  identical  with  the  problem  of  factoring  the  func- 
tion A  into  the  form  c  (.«  —  )\)  (x  —  r^)  .  .  .  (x  —  ?•„). 
Example.  — -Find  the  six  sixth  roots  of  -f  1,  and  of  —  1. 

(1)  Let  x^  =  1- 

then  ic"  -  1  =  0  =  (x^  +  1)  (x^  -  1), 

or         (x  +  1)  (x-  -  X  +  1)  (.c  -  1)  (a-  +  x  +  1)  =  0. 

This  equation  is  satisfied  when  any  factor  =  0.  Taking 
the  factors  in  order,  and  equating  to  zero,  gives  the  follow- 
ing six  roots :  — 

any  one  of  Avhich,  of  course,  taken  six  times  as  a  factor, 
makes  -j-  1. 

(2)  Let  a-«  =  _  1  ; 

then  a-«  +  1  ==  0  =  (x^  +  i)  (x^  -  i), 

or       (x  —  i)  {x-  -f  IX  —  1)  {x  +  t)  (a;^  —  /^  —  1)  =  0 ; 

whence,  as  before, 


210  NUMBER   AND   ITS   ALGEBRA. 


%Aj      V    •  *Aj      ' — -  ^"  •  t//      ~^~      €■    a  «jC      ■  ■      ■ 


'  •  rvt       ^.^        ^^       ^     «  ^y,       _^  1 * 

2  '  '  2 

any  one  of  which,  taken  six  times  as  a  factor,  makes  —  1. 

331.  In  the  application  of  JSTumber  to  concrete  problems, 
the  logic  of  the  connection  of  the  numerical  statements 
with  the  particular  concrete  conditions  must  be  thoroughly 
comprehended.  It  should  constitute  one  of  the  most  im- 
portant parts  of  mathematical  studies  and  training.  It 
ought  to  be  no  matter  for  surprise  that  numerical  results 
are  often  obtained,  totally  meaningless  in  regard  to  the 
particular  problem.  On  the  contrary,  such  results  should 
be  generally  expected,  alertly  watched  for,  in  order  to  reject 
them  from  the  problem  in  question. 

Number  is  a  twofold  continuous  magnitude,  and  there- 
fore its  thoroughgoing  application  is  possible  only  to  two- 
fold continuous  magnitudes.  {Cf.  §  188).  In  reference 
to  time,  a  one-dimensional  continuum,  all  protomonic  num- 
ber (positive  and  negative,  fractional  and  surd)  has  intel- 
ligible application ;  but  neomonic  and  complex  number 
could  have  no  application  to  temporal  relations.  To  space, 
all  number,  protomonic,  neomonic,  and  complex,  may  have 
due  application.  Sj)ace,  in  fact,  being  a  threefold  con- 
tinuum, in  a  manner  transcends  Kumber,  in  the  sense  of 
permitting  an  infinite  reapplication  of  number.  We  have 
seen,  however,  that,  given  three  planes  of  reference,  it  is 
possible  to  uniquely  determine  any  point  in  solid  space  by 
means  of  three  protomonic  numbers,  and  that  it  is  this 
circumstance  which  constitutes  the  ultimate  meaning  of 
the  statement  that  space  is  tri-dimensional. 

On  the  other  hand,  if  a  problem  require  a  number  of 
men,  it  is  limited  in  its  very  terms  to  primary  number; 
since  \  men,  or  V3  men,  would  be  as  inapplicable  as  2  -|-  7  t 


CONCRETE    PROBLEMS.  211 

men,  unless,  indeed,  implicit  reference  to  some  continuous 
magnitude  afforded  ground  for  the  application  of  such 
results ;  e.g.,  if  a  problem  concerns  the  number  of  men  in 
a  regiment,  applicable  results  are  exclusively  in  primary 
number,  and  if  such  are  not  found,  there  is  contradiction 
in  the  problem  as  given ;  whereas,  if  a  problem  concerns 
the  number  of  men  required  to  dig  a  ditch,  any  positive 
protomonic  number  might  be  interpretable. 

Not  only  must  the  student  expect  to  find  solutions  of 
his  equations  which  have  no  bearing  on  a  particular  prob- 
lem, but  it  may  be  that  no  solution  of  a  correct  algebraic 
translation  of  the  numerical  conditions  of  a  problem  is 
applicable.  The  interpretation  of  such  results  is  that 
the  problem  is  self-contradictory,  the  required  conditions 
impossible. 

The  clear  logical  principle  is,  that,  if  the  problem  have 
any  solution,  it  must  be  yielded  among  the  solutions  of 
any  system  of  algebraic  equations  which  correctly  state 
the  numerical  conditions  of  the  prolilem,  no  matter  how 
many  inapplicable  solutions  may  also  be  yielded.  If  no 
numerical  solution  is  applicable,  the  problem  is  impossible, 
that  is  to  say,  its  conditions  constitute  an  absolute  contra- 
diction of  any  such  outcome  as  was  contemplated. 

In  many  minor  ways,  also,  it  is  impossible  to  restrict 
the  perfect  generality  of  numerical  operations,  and  the 
numerical  symbols  of  the  algebra.  For  example,  an  un- 
known number  may  be  added  to  another;  but  whether  the 
addition  increases  or  decreases  a  given  number,  it  is  rash 
to  say  before  the  quality  of  the  unknown  is  discovered. 

Thus  it  is  ill-considered  to  demand  that  15  be  divided 
''  into  two  such  parts  that  the  greater  shall  exceed  3  times 
the  less  by  as  much  as  half  the  less  exceeds  three."     For 


212  NUMBER    AND    ITS    ALGEBRA. 

(representing  the  greater  by  x,  and  the  less  by  15  —  x)  the 
numerical  conditions  are  plainly  intended  to  be 

a;  _  3  (15  -  a-)  =  i  (15  -  »•)  -  3, 
whence  x  =  11,  and  15  —  a;  =  4. 

But  on  turning  to  the  requirement  it  is  seen  that  11  falls 
short  of,  not  "  exceeds  "  3  times  4  by  as  much  as  half  of  4 
falls  short  3.  In  line,  one  cannot  choose  the  issue  of  abso- 
lute facts  according  to  his  whim,  and  the  problem  as  given 
is  presumptuous  ;  all  that  could  have  been  safely  required 
were  numbers  which  would  give  equal  differences  for  the 
intended  subtractions. 

The  indeterminate  result  -  has  already  been  referred  to ; 

it  may  mean  that  any  number  answers  the  requirement,  or 
it  may  be  susceptible  of  evaluation. 

332.  Very  often  all  that  is  required  may  be  discovered 
from  equations  Avithout  solving  them,  by  transformations 
into  various  equivalent  forms.  Consequently  the  principles 
governing  the  equivalence  of  derived  and  original  systems, 
and  the  study  of  functions,  as  distinguished  from  equations, 
have,  besides  their  theoretical  importance,  a  practical  use- 
fulness quite  apart  from  their  bearing  upon  solution.  In- 
deed, the  whole  subject  of  the  solution  of  equations  has 
widened  into  that  of  the  variation  of  functions.  For  a  long 
time  equations  have  been  losing,  and  functions  gaining, 
prominence,  both  in  analytical  importance  and  practical 
utility.  Nowadays,  instead  of  seeking  merely  the  values 
of  the  variables  which  cause  the  function  to  vanish,  that  is, 
solving  the  equation  ^  (x)  =  0,  all  values  of  the  variable, 
as  it  varies  continuously,  and  the  corresponding  values  of  the 
function,  are  considered.  The  function  is  calculated  for 
enough  specific  values  of  the  variable  to  give  a  clear  idea 


VARIATION    AND   GRAPHS    OF    FUNCTIONS.         213 

of  its  variation.  Especial  attention  must  be  given  to  such 
values  of  the  variable  as  cause  the  function  to  pass  through 
critical  values,  - —  zero  among  others. 

Independently  of  the  analytical  treatment  of  geometry 
(where  the  purpose  of  geometrical  investigation  is  so 
powerfully  served  by  the  numerical  analysis),  this  modern 
way  of  regarding  analytical  functions  receives  reciprocal 
assistance  —  if  not  theoretically,  at  least  as  affording  the 
bodily  eye  a  clear  representation  —  by  drawing  what  is 
called  the  graph  of  the  function. 

The  graph  of  a  function  of  one  variable  is  plotted  by 
laying  off,  to  any  scale,  sects  proportional  to  {vide  §  213) 
arbitrarily  chosen  values  of  the  variable,  in  a  straight  line, 
to  the  riglit  or  left  of  a  point,  according  as  the  chosen 
value  is  positive  or  negative ;  and  at  the  points  so  deter- 
mined, laying  off  perpendicularly  (one  way  for  -|-,  the  other 
for  — )  sects  projjortional  to  the  corresponding  values  of 
the  function,  plotting  the  end  points  of  tiiese  sects.  By 
sketching  a  curve  through  such  points,  a  representation  of 
the  corresponding  variations  of  function  and  variable  is 
afforded.  The  curve  so  obtained  will  generally  give  warn- 
ing of  critical  values  of  the  function,  at  which  stages  closely 
consecutive  values  of  the  variable  must  be  taken  to  insure 
a  correct  graph  of  the  function. 

It  is  iisual  to  write  j/  =  4>  (-^O'  ^^^*^  ^^^^  ^^^®  values  of  y 
corresponding  to  selected  values  of  x. 

For  example,  let  the  student  plot  the  graphs  of  the 
following  functions,  also  tabulating  the  chosen  values  of  x 
with  the  corresponding  ij  's. 

(l)y  =  l-^    (2)   y=(yi^.     (3)y  =  r4i;- 


214  NUMBER   AND   ITS   ALGEBRA. 

At  first  one  may  be  disposed  to  examine  far  more  values 
than  necessary.  Always  plot  first  the  y's  corresponding 
to  X  's  which  allow  evaluation  by  inspection,  —  often  these 
will  suffice. 

A  systematic  study  of  the  variations  of  functions  would 
be  surprisingly  interesting,  even  to  students  who  have 
hitherto  found  their  mathematics  dull.  The  subject  could 
be  introduced  profitably,  even  at  very  elementary  stages  of 
algebraic  studies,  and,  while  stimulating  interest  and  sus- 
taining attention,  would  give  a  better  preparation,  both  for 
continued  study  of  pure  mathematics,  and  for  the  manifold 
I^ractical  uses  of  mathematics  in  other  sciences,  than  do  the 
methods  at  present  in  vogue. 

333.  The  general  theor^^  of  Inequalities,  and  of  Maxima 
and  Minima  values  of  functions  also,  deserves  a  more 
thorough  and  independent  treatment  than  it  commonly 
receives  in  our  elementary  text-books.  The  fundamental 
■principles  are  of  so  simple  and  instructive  a  character,  and 
form  so  valuable  an  introduction  to  the  methods  of  analysis 
employed  in  more  advanced  studies,  that  our  usual  elemen- 
tary courses  need  in  this  matter  thoroughgoing  reformation. 
The  theory  of  inequalities  is  the  best  introduction  to  that 
of  infinite  series,  and  the  latter  is  indispensable  in  the 
study  of  logarithms  and  many  other  subjects  which  are  at 
once  entered  upon  in  the  first-year  courses  of  our  colleges 
and  universities. 

For  the  most  part,  the  logic  of  inequalities,  and  the  deri- 
vation of  equivalent  inequalities,  runs  parallel  to  the 
analogous  theory  for  equations,  except  where  restrictions 
intervene  in  regard  to  inequalities,  owing  to  the  fact  that 
the  members  of  an  inequality  cannot,  like  the  members  of 
anequation,  be  interchanged. 


INEQUALITIES.  215 

The  student  may  be  reminded  (vide  §  198),  in  this  con- 
nection, that  there  is  no  comparison  in  the  ordinary  sense 
of  greater  and  less  between  complex  numbers,  because  such 
numbers  are  in  terms  of  heterogeneous  units.  Of  course 
this  general  statement  includes  particular  cases  where  one 
of  the  numbers  is  either  protomonic  or  neomonic,  and  the 
other  complex,  or  where  one  is  protomonic  and  the  other 
neomonic.  With  complex  numbers,  as  we  have  seen,  the 
comparison  must  be  between  their  moduli. 

A  fruitful  source  of  error  with  beginners  (on  account  of 
the  prevailing  inadequacy  of  number  concepts)  is  neglect 
of  the  fact  that  any  negative  number  is  less  than  zero 
(—00  <  0),  and  that  «  >  ?/,  or  x  <  y,  according  as  x  —  ?/ 
is  positive  or  negative. 

The  freedom  of  transposition  of  terms  with  changed 
signs,  in  an  inequality,  is  quite  as  immediate  a  corollary  of 
axiomatic  judgments,  and  the  significance  of  the  symbols, 
as  the  like  freedom  in  equations.  For  it  is  the  same 
axiom  that,  if  equals  be  added  to  unequals,  the  results  are 
correspondingly  unequal,  as  that,  ''if  equals  be  added  to 
equals,  the  results  are  equal."     (Vide  §   42,  foot-note.) 

Examples. 

(I.)  Prove:  x" -\- y- ':>  2  xy,  if  x  and  y  are  protomonic 
numbers,  (x  —  y)'  is  positive  whether  x  >  ?/  or  a-  <  y  ;  but 
(x  —  yY  —  cf-  —  2  xy  +  y-,  therefore  x"^  —  2  xy  -\-  y-  is  posi- 
tive, and  therefore  x-  +  y"^  >  2  xy. 

In  order  to  emphasize  the  extreme  importance  of  limit- 
ing values,  I  have  allowed  a  fallacy  to  pass  unchallenged 
in  this  argument.  It  is  not  true  that  x"^  -]-  y^  >  2  xy.  For, 
although  (x  —  ?/)-  is  positive,  it  ma}^,  if  x  =  y,  be  zero, 
when  x"^  -\-  y'^  =  2  xy  ;  consequently  the  true  statement  is 

x-  -\-  y"  not  <  2  xy. 


216  NUMBER   AND    ITS   ALGEBRA. 

(II.)  Prove:  The  sum  of  a  positive  fraction  and  its 
reciprocal  is  not  less  than  2. 

Consider  ?  + 1^  not  <  2.  (1) 

y      ^ 

Multiplying  each  member  by  ocij  gives 

a-2  +  f-  not  <  2  :nj   '  (2) 

But  (2)  has  just  been  proved  ;  therefore  its  equivalent 
inequality,  (1),  is  true. 

(III.)  Prove :  Half  the  sum  of  two  iwsitive  numbers  is 
not  less  than  the  square  root  of  their  product. 

OC     1      7/  1 

Consider  — ~^—-^  not  <  (xi/)^  (1) 

'7*"      I        '    '/*?/      I       7/ 

Squaring  gives  —  '    "  '  -^    '    "^  not  <  xy.  (2) 

But,  by  Ex.  I,  a:^  -|-  y-  not  <  2xij; 

therefore  x^  -\-  2  xy  -\-  y-  not  <  4  a-^/ ; 

therefore  (2),  and  therefore  its  equivalent  inequality  (1),  is 

true. 

This  proposition  is  readily  generalized  by  the  reasoning 
called  "  mathematical  induction,"  *  by  showing  that  if  it 
is  true  for  any  number  of  numbers,  it  is  true  for  one  more : 
—  but  it  is  true  for  two,  therefore  for  three,  and  so  on. 
Thus  we  prove  for  n  numbers  t :  — 

a-{-h  -\-c-\-   . 


n 


not  <  {(the  .  .  .)  '/'\ 


*  Not  true  and  proper  iuduction,  but  absolutely  cogent  deduction, 
involving  no  assumption  except  the  validity  of  reason,  the  postulate  of 
all  thought. 

t  Tlie  left-hand  member  is  called  the  "arithmetic  mean,"  and  the 
right,  the  "geometric  mean,"  of  the  n  numbers. 


MAXIMA   AND   MINIMA.  217 

A  maximzim  of  a  function  does  not  mean  its  greatest  pos- 
sible, nor  a  minimum  its  least  piossihle,  value.  A  maximum 
value  of  a  function  is  a  value  toward  which  it  increases, 
and  from  which  it  decreases  as  the  variable  continuously 
varies,  whether  by  increasing  or  decreasing.  And  a  mini- 
mum value  of  a  function  is  a  value  before  which  the  func- 
tion decreases,  and  after  Avhich  it  increases  as  the  variable 
varies  continuously,  whether  by  increasing  or  decreasing. 
Maxima  and  minima  for  a  function  may  repeat,  definitely 
or  indefinitely ;  or  there  may  be  only  one  maximum  or  one 
minimum  for  a  function,  in  which  case  the  maximum  is 
the  greatest  possible,  or  the  minimum  the  least  possible 
value. 

The  general  connection  between  inequalities  and  the 
theory  of  maxima  and  minima  values  of  functions  is  ex- 
emplified in  the  principle,  that  if  <^  (.y,  y,  z,  .  .  .)  and 
xj;  (x,  y,  z,  .  .  .)  be  two  functions  of  the  same  variables 
such  that 

<^  {x,  y,z,  .  .  .)=  N,  (1) 

and  xp  (x,  y,z,..   .)  not  >  <^  (.r,  y,z,   .   .   .);  (2) 

and  if  any  values  oi  x,  y,  z,  .  .  .  ,  say,  a,  h.  c,  .  .  .  ,  can  be 
found  which  satisfy  (1)  and  at  the  same  time  make  (2) 
an  equation,  then  i/^  («,  h,  c,  .  .   .)  is  a  maximum  value  of 

Also,  if  xp  (x,  y,z,  .  .  .)  =  N,  (3) 

and  </,  (.r,  y,  z,  .   .  .)  not  <  ^  {x,  y,  z,  .   .   .) ;  (4) 

and  if  any  values  oi  x,  y,  z,  .  .  .  ,  say,  a,  b,  c,  .  .  .  ,  can 
be  found  which  satisfy  (3)  and  simultaneously  make  (4) 
an  equation,  then  ^  (a,  h,  c,  .  .  .)  is  a  minimum  value  of 

Example.  —  Find  the  maximum  volume  of  a  rectangular 


218  NUMBER   AND   ITS   ALGEBRA. 

parallelopiped  of  given  surface,  and  minimum  surface  for 
given  volume. 

Let  X,  y,  and  z  be  the  lengths  of  three  adjacent  edges ; 
then  the  geometrical  data  of  the  problem  are,  the  area  of 
the  surface  is  2  (xij  -f-  ^^  -\-y^i  ^-nd  the  volume  of  the  solid 
is  xyz.  (  Vide  §  25.)  Writing  u  =  xy,  v  =  xz,  w  =  yz,  the 
area  becomes  2  (w  +  v  -)-  iv),  and  the  volume,  ^ uviv. 

Hence  the  analytical  problem  is  to  find  the  maximum 
(or  maxima)  of  the  function  -y/uviv,  given  the  function 
2  (^u  -\-  V  -\-  w)  =  a  constant.  Since  the  meaning  of  the 
problem  excludes  negative  number  (ynVZe  §  331),  the  prob- 
lem  as  assigned   is  equivalent  to  finding  the  maxima  of 

■y/uvw  given,  —  ^  —  =  k ;  for  (considering  only  posi- 
tive protomonic  numbers,  all  that  apply  to  the  problem) 
■yjuvw  is  maximum  for  the  same  values  of  the  variables 
that    ■y/uvtv    is    maximum  ;    and    given  2  (^u  -\-  v  -{-  ?/')  =  a 

constant,  we  have  —^ — — —  =  k.     But  this   transforma- 
'  3 

tion  was  adopted  because  we  know  (Ex.  Ill,  above)  that 

'  -  not  <  {iivn'Y 


o 


which  is  to  say  mat 


(iivw)''^  not  >  -— t — IL — J 
o 

Consequently  we  have 

"+;+" = k,  (1) 

and  (uvwy  not  >  !L±_L+J1'.  (2) 

o 

It  only  remains  to  find  values  of  u,  r,  ?<•  Avhich  satisfy 
(1)  and  make  (2)  an  equation.     But  (2)  cannot  be  an  equa- 


MAXIMA   AND   MINIMA.  219 

tion  unless  if  =  v  =  v\  This,  therefore,  is  the  condition, 
and  (uviv)^^  is  uniquely  maximum  when  m  =  i;  =  ?<; ;  and 
(remembering  the  meaning  of  xi,  v,  and  w),  if  u  =  v  =  w, 

then  X  =  2/  =  z  =i  —  j  ,  where  K  =  the  given  area. 

The  reciprocity,  implicit  in  the  theorem  immediately  pre- 
ceding this  example,  gives  the  same  condition  (x  =  >/  =  z) 
for  the  solution  of  the  second  part  of  this  problem ;  but  the 
beginner  may  have  failed  to  note  the  reciprocal  relations 
of  the  conditions  for  maxima  and  minima  of  two  functions 
displayed  in  the  general  investigation. 

In  like  manner,  then,  the  second  part  of  the  problem 

gives  1 

(uvwy  =  I,  (1) 

11  -A-  V  -\-  w  1 

and not  <  (uvw')^ ; 

o 

w  +  V  +  ?y  .         .  .    . 

whence, is  uniquely  minimum  when  ic  =  v  =  tv , 

and  therefore,  as  before,  the  area  is  minimum  when  x  =  y 
=  .v  =  (i)"%  where  L  is  the  given  volume. 


APPENDIX. 


PEDAGOGICAL  NOTE. 

The  primary  concept  of  number  is  the  same  in  all  men,  and  the 
conception  conld  not  be  obstructed,  even  if  teachers  set  tliemselves 
to  thwart  it.  As  an  original  question,  therefore,  there  is  little  peda- 
gogical import  in  discussion  of  methods  of  stimulating  the  infant 
mind  to  definite  specialization  of  various  manys.     (  Vide  §  2.) 

It  would  be  enough  to  jjoint  out  to  the  inexperienced  teacher  that 
when  the  time  for  definite  and  systematic  specialization  of  manys 
comes,  a  child  can  learn  the  general  system  as  a  Avhole  better  than 
he  can  learn  it  piecemeal;  that  the  so-called  arithmetic  of  the  first 
two  or  three  grades  in  our  schools  is  properly  a  matter  of  language, 
a  matter  of  naming,  in  the  manner  of  the  child's  linguistic  environ- 
ment, universal  concepts  already  attained  by  the  young  innocents 
when  committed  to  the  mercies  of  the  primary  school.  There  is  no 
more  sense  in  attempting  to  explain  icJiut  "twelve"  is,  than  in 
making  a  like  effort  in  regard  to  "time"  or  "space,"  or  such 
concepts  as  "more,"  "  less,"  "greater,"  "equal."  The  child  really 
knows  these  things  as  well  as  his  teacher.  Even  if  a  child  lived 
eight  years  in  an  English-speaking  society  Avithout  learning  the 
English  name  for  the  special  many,  "  twelve,"  or  even  without 
having  definitely  recognized  it,  the  substance  of  the  thought,  as 
distinguished  from  the  symbolism  of  a  particular  language,  would 
nevertheless  be  familiar  to  him,  and  nearly  as  well  known  as  it  can 
be  until  one  gives  profound  study  to  epistomology. 

The  simple  and  easily  taught  subjects  of  counting,  and  the  ele- 
mentary phases  of  numerical  operations,  have  been  confused  by  the 
inane  verbosity  of  pedagogical  writers.  In  his  admirable  Philos- 
ophy of  Education*  (one  of  the  best  books  ever  written  on  the 

*  Translated  in  the  International  Echication  Serks. 
221 


^ 


222  APPENDIX. 


subject)  Piosonkranz  justly  remarks,  "  Treatises  written  upon  it 
[education]  abound  more  in  sballowness  than  any  other  literature. 
Shortsightedness  and  arrogance  find  in  it  a  most  congenial  atmos- 
phere, and  uncritical  methods  and  declamatory  bombast  flourish  as 
nowhere  else." 

It  is  enough  to  point  out  one  example  of  injurious  methods  of 
dealing  with  imaginary  difficulties.  Ignorance  of  psychology  and 
lack  of  common-sense  have  led  many  superintendents,  even  where  the 
minimum  school  age  is  eight  years,  to  i:>rohibit  all  mention  of  num- 
bers greater  than  ten  in  the  "first  grade,"  and  greater  than  twenty 
in  the  "second."  This  makes  both  the  teaching  and  the  learning  a 
sham,  and  the  nemesis  of  all  dishonesty  dogs  it.  It  is  benumbing  to 
honest,  depraving  to  vain  or  deceitful,  pupils.  1  know  a  city  whose 
school  superintendent  has  instituted  such  methods  with  fatuous 
braggadocio,  where  a  visitor,  after  witnessing  an  hour's  counterfeit 
teaching,  — What  is  one  and  two?  one  and  three?  two  and  three  ? 
If  you  had  five  apples,  and  gave  one  to  Mary  and  one  to  John,  how 
many  would  you  have  left  ?  and  so  forth,  with  occasional  introduc- 
tion of  such  prodigious  numbers  as  nine  or  ten,  —  followed  the  class 
to  the  playground.  It  was  the  season  of  huUn-gnll.  Each  urchin 
knew  well  the  score  of  the  treasures  in  his  bulging  pockets.  "Iluliy- 
gull,  hand-full,  how  many  ? "  challenged  one  3'oung  plunger. 
"Twenty-two,"  guessed  his  opponent.  One  second  for  the  count 
and  the  subtraction,  and  back  came  the  triumphant  cry,  "  Give  me 
seven  to  make  it  twenty-tMo!  " 

On  the  other  hand,  there  seem  to  be  peculiar  difficulties,  even  for 
adults,  in  attaining  the  concept  of  nmnber  absolutely  essential  to 
comprehension  of  arithmetic,  —  the  discernment  of  number  as  a 
continuous  magnitude  with  fractional  parts  and  qualitative  distinc- 
tions termed  positive  and  negative.  Here  pedag6gical  devices  are 
sorely  needed.  It  is  not  enough  to  warn  against  mistaken  interfer- 
ence; the  teacher's  skill  will  be  taxed  to  the  utmost  to  stimulate  the 
minds  he  is  guiding  to  develop  concepts  of  a  high  order  of  abstrac- 
tion, and  such  as,  left  to  himself,  the  pupil  would  never  form  at  all. 

As  "object-lessons"  to  young  children  —  the  aim  being  to  clear 
up  normal  and  universal  concepts  of  quantity  —  presentation  of 
yard-sticks  and  foot-rules,  gallon  and  quart  measures,  etc.,  may  be 
ii  useful  practice,  and  it  does  teach  about  fractions  ;  but  it  does  not 


APPENDIX.  223 

at  all  immediately  suggest  fractions  of  numbers.  A  fraction  of  a 
line  is  a  line,  of  a  solid  is  a  solid  ;  and  these  can  be  and  universally 
are  discerned  under  the  primary  concept  of  number,  and  without 
discernment  of  numerical  fractions.  Every  savage  knows  that  a 
quart  is  a  fraction  of  a  gallon.  The  "object-lessons"  mentioned 
really  constitute  an  elementary  discipline  in  geometry  (if  every 
primary  school  exercise  must  be  labelled  with  the  name  of  some 
science).  Lines  and  solids  are  spacial  entities,  and  contempla- 
tion of  their  relations  is  primarily  a  geometrical  exercise.  I  say 
X>riniarili/,  because  any  two  magnitudes  of  the  same  kind  have  an 
absolute  numerical  relation  ;  but  to  see  that  a  quart  is  one-fourth  of 
a  gallon  (only  another  way  of  saying  that  four  quarts  equal  a 
gallon)  is  not  at  all  to  see  the  number  called  one-fourth  in  the 
systematic  terminology  of  arithmetic.  Every  child  sees  the  former, 
an  obvious  geometric  fact,  —  too  many  of  his  teachers  have  never 
discerned  the  latter.  {Vide  §§  TS-90.)  The  primary  concept  of 
number  is  universal  and  normal  to  the  human  mind,  just  as  the 
concei^t  of  space  is  common  and  original  to  all  men.  Systematic 
development  of  the  latter  gives  geometry,  of  the  former  gives  arith- 
metic. The  developments  of  the  one  are  quite  as  much  matters  of 
fact  as  the  developments  of  the  other. 

Ontological  definition  of  number  is  as  little  to  be  required  of 
arithmetic  as  like  definitions  of  space  of  geometry,  or  of  matter  and 
force  of  physics.  Each  science  simply  takes  its  respective  common 
notions,  which  it  develops  according  to  inherent  characteristics. 
The  developed  science  always  casts  light  back  upon  primary  notions 
(Cy.  the  effect  of  Xon-Euclidean  geometries  upon  native  ideas  of 
space,  or  the  exigencies  of  dilemmas  in  physics  upon  naive  con- 
cepts of  matter) ;  but  no  such  questions  are  to  be  raised  for  young 
students  beginning  to  study  arithmetic,  geometry,  or  physics.  The 
most  important  maxim  for  wise  teaching  in  any  science  is  never  to 
set  delimitations  which  confine  development  and  entomb  thought  in 
empiricism,  —  never  to  clip  the  growing  tree  at  the  top. 

Now  every  man  (and  every  dog)  knows  that  one  side  of  a  triangle 
is  less  than  the  sum  of  the  other  two  sides  ;  but  no  one  would  sup- 
pose that  this  circumstance  entitled  every  man  to  opinions  concern- 
ing the  conclusions  of  geometry  ;  yet  similar  presumptions  are  rife 
among  teachers  of  arithmetic.     Men  possessing  (in  common  with 


224  APPENDIX. 

their  most  savage  brethren)  only  the  primary  concept  with  which 
arithmetic  begins,  often  misrepresent  as  matter  of  convention  or 
symbolic  jugglery  the  arithmetical  conclusions  that  number  is  a 
continuous  magnitude,  with  fractional  parts,  and  qualitative  dis- 
tinctions — •  as  much  matters  of  fact  as  any  conclusions  of  geometry. 

The  developments  of  the  number  concept  are  undreamed  of  to  the 
man  whose  only  thought  thereof  is  his  abstraction  from  a  flock  of 
sheep  or  pile  of  coins.  As  soon  as  man's  energetic  and  organizing 
thought  develops  this  concept,  the  insight  is  infallibly  attained  that 
number  is  a  continuous  magnitude,  not  concrete,  not  material,  but 
none  the  less  real. 

The  concept  which  appears  to  me  most  like  the  first  development 
of  primaiy  number,  which  includes  all  ratio  (including  fractions), 
and  the  qualitative  distinctions,  positive  and  negative,  is  Time. 
Even  children  recognize  time  as  a  continuous  magnitude,  —  as  more 
or  less;  that  of  two  times  one  must  be  definitely  greater  than,  equal 
to,  or  less  than  the  other,- — and  the  qualitative  distinctions  of  i^ast 
and  future.     The  analogy  of  jjresejii  and  zero  is  also  perfect. 

I  suggest  that  teachers,  called  upon  as  they  always  are  to  teach 
arithmetic  to  children  somewhat  too  3'oung  for  the  reasoning  and 
insight  required,  would  do  well,  in  attempting  to  stimulate  the  con- 
ceptual energies  of  their  pupils,  to  use  definite  times  rather  than 
lines,  surfaces,  solids,  etc.,  in  illustrating  numerical  relations  sub- 
sisting between  any  two  magnitudes  of  the  same  kind.  Altliough 
no  better  success  can  be  assured  in  this  way  (for  any  fraction  or  part 
of  a  time  is  a  time  and  not  a  number);  yet  from  the  very  fact  that 
times  cannot  be  seen  or  handled,  the  abstracting  functions  of  the 
mind  are  brought  into  play,  and  there  is  better  ground  of  hope  that 
the  desired  conception  will  take  place  than  if  objects  of  sense- 
perception  had  been  presented.  It  may  be  well  to  remark  in  this 
connection  that  in  all  illustration  great  care  is  demanded  lest  the 
analog  hide  instead  of  revealing.  Rosenkranz,  in  his  valuable  Phi- 
loHophy  of  Education,  already  referred  to,  wisely  cautions  :  "  Our  age 
inclines  at  present  to  the  superstition  that  man  is  able,  by  means  of 
simple  sense-perception,  to  attain  a  knowledge  of  the  essence  of 
things,  and  thereby  dispense  with  the  trouble  of  thinking.  It  is 
vain  to  try  to  get  behind  things,  or  to  comprehend  them,  except  by 
thinking," 


APPENDIX.  225 

I  am  not  aware  that  the  suggestion  has  been  made  hitherto  ;  but, 
in  the  Hght  of  tlie  above  warning  against  abuse,  1  am  convinced  that 
teachers  of  arithmetic  would  do  well  to  contemplate  the  similarity 
of  the  concepts  Time  and  Number,  as  the  latter  is  conceived,  not  in 
the  savage  stadium  of  thought,  but  in  its  first  scientific  develop- 
ment. 

There  are  many  subsidiary  advantages  also  in  choosing  the  uni- 
versally conceived  magnitude,  time,  for  such  illustrations.  The 
mind  is  unconsciously  but  directly  led  from  the  tyranny  of  material 
categories  of  thought;  and  the  human  mind,  once  made  sensible  of 
its  powers,  will  never  again  suffer  its  conceiJtions  to  be  shackled  in 
this  native  slavery  of  the  race. 

Rightly  employed,  arithmetic  might  be  used  with  more  efficacy  in 
the  intellectual  emancipation,  which  is  one  of  the  chief  ends  of  edu- 
cation, than  any  subject  in  the  curricula  of  common  schools.  There 
is  no  other  field  where  one  pure  idea  is  developed  in  such  unbroken 
consistency,  and  such  freedom  from  involvement  in  complex  rela- 
tions with  foreign  elements. 

In  conclusion,  no  matter  whether  the  pupil  at  a  given  stage  be  in 
a  position  to  see  the  end  of  his  studies  or  not,  it  is  evident  that  the 
teacher,  with  no  notion  of  the  end,  will  be  a  faulty  guide,  since  he 
leads  he  knows  not  whither. 


INDEX. 


The  numbers  refer  to  sections. 


Addition,  Si,  37,  45-,  73, 88, 162, 184, 
199. 

Algebra,  20-,  32,  71,  15(3,  192,  224, 
236,  251. 

Algebraic  Form :  vide  Form. 

Analytics,  27,  169,  234. 

Angles,  211,  212. 

Arithmetic,  14,  30-32,  71,  10(5,  192. 

Arithmetic,  Pure  and  Applied,  31. 

Association,  Laws  of,  39,  43,  46,  47, 
51,  62,  71,  73. 

Axioms,  42,  333. 

Base  of  Notation :  cf.  Radix,  Scales, 
7,  17. 

Billion,  8. 

Calculation,  29,  95,  210. 

Calculus,  130,  132,  222. 

Cardinal  Numbers,  10. 

Circulating  Decimals:  vide  Re- 
peating. 

Coefficients,  Theorem  of  Undeter- 
mined, 267-268. 

Commensurable,  83,  145,  174,  205. 

Commutative  Laws,  34,  38,  43,  46, 
47,  51,  62,  65,  73. 

Complex  Number,  27,  145,  180, 
186-,  193-,  291. 

Comijosition  of  Ratios,  84. 

Computation,  Devices  of,  72,  93,  95. 

Concept,  Number:  vide  Number 
Concept. 

Concepts,  Elemental  Mathemati- 
cal, 222,  228-. 


Concrete  Problems,  27,  30,  48,  210, 

331. 
Congruence,  298.        *■ 
Continuity  of  Number,  80-82,  97, 

188,  198. 
Continuity,     Principle     of:     vide 

Principle. 
Counting,  5,  9,  12,  14. 
Cube :  Cube  Root,  58,  76,  249,  290. 
Decimal  Fractions:  cf.  Radix,  17. 
Decimal  Notation  :  cf.  Notation,  17. 
Definitions :  cf.  respective  heads :  — 

Addition,  37. 

Algebra,  20. 

Arithmetic,  32. 

Calculation,  29. 

Commensurable,  83. 

Counting,  5. 

Division,  49. 

Evolution,  GG,  68. 

Finding  Logarithm,  G9. 

Fraction,  S3. 

Incomtnensurablc,  83. 

Inrolution,  (50. 

Mathematics,  225. 

Multiple,  83. 

Multiplication,  40. 

Notation,  14. 

Primary  Number,  2,  3. 

Proportional,  213. 

Patio,  83. 

Submultiple,  83. 

Subtraction,  41,  42. 

Surd,  83. 

Etc. 


22*7 


228 


INDEX. 


Degree,  169,  323. 
Denominator,  87. 
Dialectic,  11,  4.5,  W,  80-,  110,  113- 

114,  181,  202,  230. 
Dimensions,  188,  231. 
Discrete,  2,  80-81,  97,  229. 
Distributive  Law,  52-,  73. 
Division,  49,  73,  84,  89-,  90,  98, 123, 

103,  184,  199. 
Division,  Algebraic,  254-. 
Duodecimals,  17,  282-283. 
Enumeration,  5. 
Equation,  Synthetic,  40. 
Equations,  — 

Classification  of,  299. 
Equivalence  of  :    vide    Equiva- 
lence. 
Higher:    cf.  Equations,   Theory 

of,  330. 
Indeterminate,  317. 
Quadratic,  313-. 
Roots  of :  vide  Hoots. 
Simultaneous,  300,  317,  327,  330. 
Solutions  of:  vide  Solutions. 
Systems  of:  vide  Equivalence. 
Theory  of,  2G8-,  30.3-. 
Transformation  of,  309-,  .332. 
Equivalence  of  Equations,  319-. 
Evolution,  34,  68,  76,  92,  98,   143, 

163. 
Exponential  Notation,  57,  146,  150, 

156. 
Exponents,  Law  of,  64,    146,    158, 

191. 
Factors,  Algebraic :  cf.  Hiyhest  and 
Theory  of  Equations,  251,  304. 
Finding  Logarithm,  34,  69. 
Form,  Algebraic,  28,  156,  2.36,  251, 

301. 
Formula,  40,  299. 

Formulae  of  Definition,  42,  49,  68, 
69. 


Fractions,  78-,  80-85,  89-,  143,  278, 
285. 

Functions,  — 

Classification  of,  169,  263,  265. 
Variation  of,  332. 
Graphs  of,  332. 

Fundamental   Theory:    vide    The- 
ory. 

Geometry,  22,  25,  181,  193,  227,  230, 
2;!5,  297. 

Graph  of  Function,  332. 

Greater,  11,  46,  116,  198,  240,  242, 
252,  333. 

Greatest   Common  Measure:  vide 
Submultiple. 

Highest    Common    Factor,    Alge- 
braic, 251,  272. 

Homogeneous  Functions,  263. 

Homogeneous  Manifoldness,  229. 

Identity:  vide  Formula. 

"Imaginary:"  cf.   Neomonic,   99, 
101,  145,  181,  191. 

Incommensurable,  83,  95,  145,  174, 
205. 

Indeterminate     Equations:     vide 
Equations. 

Indeterminate  Forms,  130-,  135-. 

Indices,  Law  of,  64,  146,  158,  191. 

Inequalities,  3.33. 

Infinitesimals,  222. 

Infinity,  133-,  222. 

Integers,  1,  241. 

Involution,  34,  59,  60,  75,  92,  163, 
184. 

"  Irrational  "  :  (/.  Surd  and  Incom- 
mensurable, 104,  145,  181. 

Less,  11,  46,  116,  198,  240,  242,  252, 
333. 

Logarithm,     Finding     the :      vide 
Finding  the  Logarithm. 

Logarithms,  57,  150-,  333. 

Lo"-ic   21    222 


INDEX. 


229 


Lowest   Conimou   Multiple  :    vide 

Multiple. 
Lowest  Common    Multiple,  Alge- 
braic, 251,  276. 
Magnitude,  11,  207,  210,  229. 
Manifoldness,  229-. 
Many,  2,  228. 
Manys,  Specialized,  2,  37. 
Mathematics,  105,  222-,  234. 
Maxima,  333. 
Measure,  204,  205,  209. 
Measurement,  12,  80,  203,  211,  227. 
INIensuration,    203-,    211,    220,    227, 

231. 
Metre,  220. 

Metric  System,  31,  220. 
Minima,  333. 

Minus,  Double  Meaning  of-,  120. 
Modulus    of    Complex    Numbers, 
198,  293,  333. 
Congruence,  298. 
Logarithms,  150. 
Multiple,  83. 

Multiple,  Lowest  Common,  242. 
Multiplication,    34,  44-48,   73,   89, 

121,  163,  184,  199. 
Negative  Number,  27,  99,  104,  110, 

117,  143,  181,  333. 
Neomon,  26,  104,  181,  182,  191. 
Neomonic    Number,  27,   104,   145i 

181-  191. 
Nine,  Remainders  to,  287-. 
Nines,    Casting  out :    vide    Nine, 

Eemainders  to. 
Norm  of  Complex  Numbers,  197, 

293. 
Notation,   14,   17,   57,  90,   120,   150, 

156,  277-,  280-,  284-. 
Notation,  History  of,  16. 
Number  Concept,  2-,  27,  34,71,  78-, 

80,  84-,  96-115,  155,  179-,  186, 

192,  202,  226,  230. 


Number,  — 
Development  of  :    vide  Number 

Concept. 
Origin  of  :  vide  Origin. 
Primary,  Fractional,  Surd,  Posi- 
tive and  Negative,  Neomomic, 

Complex  :  vide  Corresponding 

heads. 
Numbers,  Theory  of  :  vide  Theortj 

of  Numbers. 
Numerals,  6,  8,  15. 
Numeration,  5,  7,  17. 
Numerator,  87. 
One,  2,  128,  181. 
Operations,  30,    34,  45-,  60.  73,  86, 

104,     116,    130-,    135-184,    199, 

210. 
Ordinals,  10. 

Origin  of  Number,  2,  5,  2.'50. 
Physics,  21,  27,  211,  221,  224. 
Plus,  Double  Meaning  of  +,  120. 
Primary  Number,  1-4,   11,  54,  60. 

70,  96,  181. 
Prime  Numbers,  238-. 
Primeness,  Algebraic,  251,  275-. 
Principle    of    Continuity,  61,  97-, 

103-,  107,  155. 
Projective  Geometry,  227. 
Proportionality,  211-219. 
Protomonic  Number,  101,  145,  154. 
Quadratics  :  vide  Equations. 
Quality,  14,  27,  32,  78, 110,  117,  120, 

224,  228. 
Quantity,  198,  224,  228-. 
Radix  Fractions,  17,  284-. 
Radix  Notations,  17,  280-. 
Radicals,     Radical-Surds,    83,    94, 

145,  157-,  170-,  249. 
Ratio,  25,  78-,  83,  84,  132,  203,  208. 
Reciprocal,  90. 
Remainder,  Least,  243,  255. 
Remainder  Theorem,  258-. 


230 


INDEX. 


Remainder  to  Nine,  287-. 
Repeating  Decimals,  249,  284-. 
Roots  of  Equations,  268,  271,  300, 

306-,  315,  323. 
Rules,  43,  88,  93,  119,  121,  124,  242. 
Scales,  Notational,  277-. 
Series,  267,  333. 
Solutions    of    Equations,   300-302, 

331. 
Square  :    Square  Root,  58,  76,  93-, 

191,  200,  249,  290. 
Stirpal,  145. 

Submultiple,  83,  84,  205,  251. 
Submultiple,    Highest    Common, 

205,  240,  251. 
Subtraction,  34,  41-,  45,  89,  98,  117, 

162,  199. 
Surds,  80,  83,  94,  143,  145,  174,  249. 
Symbols,  14,  17,  20,  23,  32,  192. 
Symmetrical  Functions,  265. 


Synoptic  Mathematical  Methods, 

234. 
Synthetic  Mathematical  Methods : 

vide  Synoptic. 
Synthetic  Equation,  40. 
Systems :     vide     Equivalence    of 

Equations,  and  Manifoldness. 
Terminology,  83,  101,  145,  198,  205. 
Theory,  Fundamental,  2, 11,  26,  30, 

71,  97-,  107,  132,  207,  222,   225, 

228,  230,  331. 
Theory  of  Equations,  268-,  303-. 
Theory  of  Numbers,  298. 
Undetermined    Coefficients,    267- 

268. 
Unit,  203-206,  220. 
Unity,  2,  16,  128,  181,  203-. 
Variation  of  Functions,  332. 
Variables,  299. 
Zero,  16,  116,  125-,  182,  189,  222. 


ERRATA. 


Page  52. 

2d.  line  from  bottom  :  instead  oi  b  —  d  read  h  j d. 

Page  83. 

Top  line  :  instead  of  "  point  to  the  vertices  "  read  points 
to  the  ojjj^osite  vertices. 

Page  105. 

13th  line  from  bottom :   sign  of  equality  is  omitted  in 
latter  portion  of  the  line. 

Page  115. 

7th  line  from  bottom :  read  §  198. 

Page  165. 

3d  line  :  read  x  —  y  in  denominator. 

Page  174. 

11th  line:  read  c?3r~^ 

Page  ISO. 

7th  line:  read^  +^. 
r         v 

Page  203. 

11th  line  :  instead  of  1  —  6  in  second  denominates-,  read 
1-x. 


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LIBRARY  USE 

RETURN  TO  DBSK  FROM  WHICH  BORROWED 

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